Integrate x^2(x-4)^0.5

\int x^2\sqrt{x-4}dx

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Answer

$-256\frac{\frac{1}{160}\sqrt{x-4}x^{5}}{x^2}+256\frac{\frac{1}{896}\sqrt{x-4}x^{7}}{x^2}-68.2667\sqrt{x-4}-204.8\frac{\frac{1}{24}\sqrt{x-4}x^{3}}{x^2}+219.4286\frac{\frac{1}{160}\sqrt{x-4}x^{5}}{x^2}+58.5143\sqrt{x-4}+175.5429\frac{\frac{1}{24}\sqrt{x-4}x^{3}}{x^2}+C_0$

Step by step solution

Problem

$\int x^2\sqrt{x-4}dx$
1

Solve the integral $\int\sqrt{x-4}x^2$ by trigonometric substitution using the substitution

$\begin{matrix}x=4\sec\left(\theta\right)^{2} \\ dx=8\tan\left(\theta\right)\sec\left(\theta\right)^2d\theta\end{matrix}$
2

Substituting in the original integral, we get

$\int128\tan\left(\theta\right)\sec\left(\theta\right)^2\sqrt{4\sec\left(\theta\right)^{2}-4}\sec\left(\theta\right)^{4}d\theta$
3

Taking the constant out of the integral

$128\int\tan\left(\theta\right)\sec\left(\theta\right)^2\sqrt{4\sec\left(\theta\right)^{2}-4}\sec\left(\theta\right)^{4}d\theta$
4

Factor by the greatest common divisor $4$

$128\int\tan\left(\theta\right)\sec\left(\theta\right)^2\sqrt{4\left(\sec\left(\theta\right)^{2}-1\right)}\sec\left(\theta\right)^{4}d\theta$
5

The power of a product is equal to the product of it's factors raised to the same power

$128\int2\tan\left(\theta\right)\sec\left(\theta\right)^2\sec\left(\theta\right)^{4}\sqrt{\sec\left(\theta\right)^{2}-1}d\theta$
6

When multiplying exponents with same base we can add the exponents

$128\int2\tan\left(\theta\right)\sqrt{\sec\left(\theta\right)^{2}-1}\sec\left(\theta\right)^{6}d\theta$
7

Applying the trigonometric identity: $\tan\left(\theta\right)^2=\sec\left(\theta\right)^2-1$

$128\int2\tan\left(\theta\right)\tan\left(\theta\right)\sec\left(\theta\right)^{6}d\theta$
8

When multiplying exponents with same base you can add the exponents

$128\int2\sec\left(\theta\right)^{6}\tan\left(\theta\right)^2d\theta$
9

Taking the constant out of the integral

$128\cdot 2\int\sec\left(\theta\right)^{6}\tan\left(\theta\right)^2d\theta$
10

Multiply $2$ times $128$

$256\int\sec\left(\theta\right)^{6}\tan\left(\theta\right)^2d\theta$
11

Applying the trigonometric identity: $\tan^2(\theta)=\sec(\theta)^2-1$

$256\int\sec\left(\theta\right)^{6}\left(\sec\left(\theta\right)^2-1\right)d\theta$
12

Multiplying polynomials $\sec\left(\theta\right)^{6}$ and $\sec\left(\theta\right)^2+-1$

$256\int\left(\sec\left(\theta\right)^{8}-\sec\left(\theta\right)^{6}\right)d\theta$
13

The integral of a sum of two or more functions is equal to the sum of their integrals

$256\left(\int-\sec\left(\theta\right)^{6}d\theta+\int\sec\left(\theta\right)^{8}d\theta\right)$
14

Taking the constant out of the integral

$256\left(\int\sec\left(\theta\right)^{8}d\theta-\int\sec\left(\theta\right)^{6}d\theta\right)$
15

Simplify the integral of secant applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$

$256\left(\int\sec\left(\theta\right)^{8}d\theta-\left(\frac{4}{5}\int\sec\left(\theta\right)^{4}d\theta+\frac{\sec\left(\theta\right)^{5}\sin\left(\theta\right)}{5}\right)\right)$
16

Expressing the result of the integral in terms of the original variable

$256\left(\int\sec\left(\theta\right)^{8}d\theta-\left(\frac{4}{5}\int\sec\left(\theta\right)^{4}d\theta+\frac{\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x}}{5}\right)\right)$
17

Simplifying the fraction

$256\left(\int\sec\left(\theta\right)^{8}d\theta-\left(\frac{4}{5}\int\sec\left(\theta\right)^{4}d\theta+\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{5x}\right)\right)$
18

Taking out the constant $5$ from the fraction's denominator

$256\left(\int\sec\left(\theta\right)^{8}d\theta-\left(\frac{4}{5}\int\sec\left(\theta\right)^{4}d\theta+\frac{\frac{1}{5}\cdot\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x}}{x}\right)\right)$
19

Simplify the integral of secant applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$

$256\left(\int\sec\left(\theta\right)^{8}d\theta-\left(\frac{4}{5}\left(\frac{2}{3}\int\sec\left(\theta\right)^{2}d\theta+\frac{\sec\left(\theta\right)^{3}\sin\left(\theta\right)}{3}\right)+\frac{\frac{1}{5}\cdot\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x}}{x}\right)\right)$
20

Expressing the result of the integral in terms of the original variable

$256\left(\int\sec\left(\theta\right)^{8}d\theta-\left(\frac{4}{5}\left(\frac{2}{3}\int\sec\left(\theta\right)^{2}d\theta+\frac{\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x}}{3}\right)+\frac{\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x}}{x}\right)\right)$
21

Simplifying the fraction

$256\left(\int\sec\left(\theta\right)^{8}d\theta-\left(\frac{4}{5}\left(\frac{2}{3}\int\sec\left(\theta\right)^{2}d\theta+\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{3x}\right)+\frac{\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x}}{x}\right)\right)$
22

Simplifying the fraction

$256\left(\int\sec\left(\theta\right)^{8}d\theta-\left(\frac{4}{5}\left(\frac{2}{3}\int\sec\left(\theta\right)^{2}d\theta+\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{3x}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x\cdot x}\right)\right)$
23

When multiplying exponents with same base you can add the exponents

$256\left(\int\sec\left(\theta\right)^{8}d\theta-\left(\frac{4}{5}\left(\frac{2}{3}\int\sec\left(\theta\right)^{2}d\theta+\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{3x}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)\right)$
24

Taking out the constant $3$ from the fraction's denominator

$256\left(\int\sec\left(\theta\right)^{8}d\theta-\left(\frac{4}{5}\left(\frac{2}{3}\int\sec\left(\theta\right)^{2}d\theta+\frac{\frac{1}{3}\cdot\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x}}{x}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)\right)$
25

The integral of $\sec(x)^2$ is $\tan(x)$

$256\left(\int\sec\left(\theta\right)^{8}d\theta-\left(\frac{4}{5}\left(\frac{2}{3}\tan\left(\theta\right)+\frac{\frac{1}{3}\cdot\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x}}{x}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)\right)$
26

Expressing the result of the integral in terms of the original variable

$256\left(\int\sec\left(\theta\right)^{8}d\theta-\left(\frac{4}{5}\left(\frac{2}{3}\cdot\frac{\sqrt{x-4}}{2}+\frac{\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x}}{x}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)\right)$
27

Simplify the fraction

$256\left(\int\sec\left(\theta\right)^{8}d\theta-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x}}{x}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)\right)$
28

Simplifying the fraction

$256\left(\int\sec\left(\theta\right)^{8}d\theta-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x\cdot x}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)\right)$
29

When multiplying exponents with same base you can add the exponents

$256\left(\int\sec\left(\theta\right)^{8}d\theta-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)\right)$
30

Simplify the integral of secant applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\int\sec\left(\theta\right)^{6}d\theta+\frac{\sec\left(\theta\right)^{7}\sin\left(\theta\right)}{7}\right)$
31

Expressing the result of the integral in terms of the original variable

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\int\sec\left(\theta\right)^{6}d\theta+\frac{\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x}}{7}\right)$
32

Simplifying the fraction

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\int\sec\left(\theta\right)^{6}d\theta+\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{7x}\right)$
33

Taking out the constant $7$ from the fraction's denominator

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\int\sec\left(\theta\right)^{6}d\theta+\frac{\frac{1}{7}\cdot\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x}}{x}\right)$
34

Simplify the integral of secant applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\left(\frac{4}{5}\int\sec\left(\theta\right)^{4}d\theta+\frac{\sec\left(\theta\right)^{5}\sin\left(\theta\right)}{5}\right)+\frac{\frac{1}{7}\cdot\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x}}{x}\right)$
35

Expressing the result of the integral in terms of the original variable

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\left(\frac{4}{5}\int\sec\left(\theta\right)^{4}d\theta+\frac{\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x}}{5}\right)+\frac{\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x}}{x}\right)$
36

Simplifying the fraction

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\left(\frac{4}{5}\int\sec\left(\theta\right)^{4}d\theta+\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{5x}\right)+\frac{\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x}}{x}\right)$
37

Simplifying the fraction

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\left(\frac{4}{5}\int\sec\left(\theta\right)^{4}d\theta+\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{5x}\right)+\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x\cdot x}\right)$
38

When multiplying exponents with same base you can add the exponents

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\left(\frac{4}{5}\int\sec\left(\theta\right)^{4}d\theta+\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{5x}\right)+\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x^2}\right)$
39

Taking out the constant $5$ from the fraction's denominator

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\left(\frac{4}{5}\int\sec\left(\theta\right)^{4}d\theta+\frac{\frac{1}{5}\cdot\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x}}{x}\right)+\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x^2}\right)$
40

Simplify the integral of secant applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\left(\frac{4}{5}\left(\frac{2}{3}\int\sec\left(\theta\right)^{2}d\theta+\frac{\sec\left(\theta\right)^{3}\sin\left(\theta\right)}{3}\right)+\frac{\frac{1}{5}\cdot\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x}}{x}\right)+\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x^2}\right)$
41

Expressing the result of the integral in terms of the original variable

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\left(\frac{4}{5}\left(\frac{2}{3}\int\sec\left(\theta\right)^{2}d\theta+\frac{\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x}}{3}\right)+\frac{\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x}}{x}\right)+\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x^2}\right)$
42

Simplifying the fraction

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\left(\frac{4}{5}\left(\frac{2}{3}\int\sec\left(\theta\right)^{2}d\theta+\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{3x}\right)+\frac{\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x}}{x}\right)+\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x^2}\right)$
43

Simplifying the fraction

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\left(\frac{4}{5}\left(\frac{2}{3}\int\sec\left(\theta\right)^{2}d\theta+\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{3x}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x\cdot x}\right)+\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x^2}\right)$
44

When multiplying exponents with same base you can add the exponents

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\left(\frac{4}{5}\left(\frac{2}{3}\int\sec\left(\theta\right)^{2}d\theta+\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{3x}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x^2}\right)$
45

Taking out the constant $3$ from the fraction's denominator

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\left(\frac{4}{5}\left(\frac{2}{3}\int\sec\left(\theta\right)^{2}d\theta+\frac{\frac{1}{3}\cdot\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x}}{x}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x^2}\right)$
46

The integral of $\sec(x)^2$ is $\tan(x)$

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\left(\frac{4}{5}\left(\frac{2}{3}\tan\left(\theta\right)+\frac{\frac{1}{3}\cdot\frac{\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x}}{x}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x^2}\right)$
47

Expressing the result of the integral in terms of the original variable

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\left(\frac{4}{5}\left(\frac{2}{3}\cdot\frac{\sqrt{x-4}}{2}+\frac{\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x}}{x}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x^2}\right)$
48

Simplify the fraction

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x}}{x}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x^2}\right)$
49

Simplifying the fraction

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x\cdot x}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x^2}\right)$
50

When multiplying exponents with same base you can add the exponents

$256\left(-\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{6}{7}\left(\frac{4}{5}\left(\frac{1}{3}\sqrt{x-4}+\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}\right)+\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x^2}\right)$
51

Multiply $\left(\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}+\frac{1}{3}\sqrt{x-4}\right)$ by $175.5429$

$-256\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}+256\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x^2}-68.2667\sqrt{x-4}-204.8\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}+219.4286\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}+58.5143\sqrt{x-4}+175.5429\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}$
52

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$-256\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x^{5}}{32}\right)}{x^2}+256\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x^2}-68.2667\sqrt{x-4}-204.8\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}+219.4286\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}+58.5143\sqrt{x-4}+175.5429\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}$
53

Simplify the fraction

$-256\frac{\frac{1}{160}\sqrt{x-4}x^{5}}{x^2}+256\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x}{2}\right)^{7}}{x^2}-68.2667\sqrt{x-4}-204.8\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}+219.4286\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}+58.5143\sqrt{x-4}+175.5429\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}$
54

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$-256\frac{\frac{1}{160}\sqrt{x-4}x^{5}}{x^2}+256\frac{\frac{1}{7}\sqrt{x-4}\left(\frac{x^{7}}{128}\right)}{x^2}-68.2667\sqrt{x-4}-204.8\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}+219.4286\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}+58.5143\sqrt{x-4}+175.5429\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}$
55

Simplify the fraction

$-256\frac{\frac{1}{160}\sqrt{x-4}x^{5}}{x^2}+256\frac{\frac{1}{896}\sqrt{x-4}x^{7}}{x^2}-68.2667\sqrt{x-4}-204.8\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}+219.4286\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}+58.5143\sqrt{x-4}+175.5429\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}$
56

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$-256\frac{\frac{1}{160}\sqrt{x-4}x^{5}}{x^2}+256\frac{\frac{1}{896}\sqrt{x-4}x^{7}}{x^2}-68.2667\sqrt{x-4}-204.8\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x^{3}}{8}\right)}{x^2}+219.4286\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}+58.5143\sqrt{x-4}+175.5429\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}$
57

Simplify the fraction

$-256\frac{\frac{1}{160}\sqrt{x-4}x^{5}}{x^2}+256\frac{\frac{1}{896}\sqrt{x-4}x^{7}}{x^2}-68.2667\sqrt{x-4}-204.8\frac{\frac{1}{24}\sqrt{x-4}x^{3}}{x^2}+219.4286\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x}{2}\right)^{5}}{x^2}+58.5143\sqrt{x-4}+175.5429\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}$
58

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$-256\frac{\frac{1}{160}\sqrt{x-4}x^{5}}{x^2}+256\frac{\frac{1}{896}\sqrt{x-4}x^{7}}{x^2}-68.2667\sqrt{x-4}-204.8\frac{\frac{1}{24}\sqrt{x-4}x^{3}}{x^2}+219.4286\frac{\frac{1}{5}\sqrt{x-4}\left(\frac{x^{5}}{32}\right)}{x^2}+58.5143\sqrt{x-4}+175.5429\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}$
59

Simplify the fraction

$-256\frac{\frac{1}{160}\sqrt{x-4}x^{5}}{x^2}+256\frac{\frac{1}{896}\sqrt{x-4}x^{7}}{x^2}-68.2667\sqrt{x-4}-204.8\frac{\frac{1}{24}\sqrt{x-4}x^{3}}{x^2}+219.4286\frac{\frac{1}{160}\sqrt{x-4}x^{5}}{x^2}+58.5143\sqrt{x-4}+175.5429\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x}{2}\right)^{3}}{x^2}$
60

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$-256\frac{\frac{1}{160}\sqrt{x-4}x^{5}}{x^2}+256\frac{\frac{1}{896}\sqrt{x-4}x^{7}}{x^2}-68.2667\sqrt{x-4}-204.8\frac{\frac{1}{24}\sqrt{x-4}x^{3}}{x^2}+219.4286\frac{\frac{1}{160}\sqrt{x-4}x^{5}}{x^2}+58.5143\sqrt{x-4}+175.5429\frac{\frac{1}{3}\sqrt{x-4}\left(\frac{x^{3}}{8}\right)}{x^2}$
61

Simplify the fraction

$-256\frac{\frac{1}{160}\sqrt{x-4}x^{5}}{x^2}+256\frac{\frac{1}{896}\sqrt{x-4}x^{7}}{x^2}-68.2667\sqrt{x-4}-204.8\frac{\frac{1}{24}\sqrt{x-4}x^{3}}{x^2}+219.4286\frac{\frac{1}{160}\sqrt{x-4}x^{5}}{x^2}+58.5143\sqrt{x-4}+175.5429\frac{\frac{1}{24}\sqrt{x-4}x^{3}}{x^2}$
62

Add the constant of integration

$-256\frac{\frac{1}{160}\sqrt{x-4}x^{5}}{x^2}+256\frac{\frac{1}{896}\sqrt{x-4}x^{7}}{x^2}-68.2667\sqrt{x-4}-204.8\frac{\frac{1}{24}\sqrt{x-4}x^{3}}{x^2}+219.4286\frac{\frac{1}{160}\sqrt{x-4}x^{5}}{x^2}+58.5143\sqrt{x-4}+175.5429\frac{\frac{1}{24}\sqrt{x-4}x^{3}}{x^2}+C_0$

Answer

$-256\frac{\frac{1}{160}\sqrt{x-4}x^{5}}{x^2}+256\frac{\frac{1}{896}\sqrt{x-4}x^{7}}{x^2}-68.2667\sqrt{x-4}-204.8\frac{\frac{1}{24}\sqrt{x-4}x^{3}}{x^2}+219.4286\frac{\frac{1}{160}\sqrt{x-4}x^{5}}{x^2}+58.5143\sqrt{x-4}+175.5429\frac{\frac{1}{24}\sqrt{x-4}x^{3}}{x^2}+C_0$

Problem Analysis

Main topic:

Integration by trigonometric substitution

Time to solve it:

3.15 seconds

Views:

74