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Integral of (x^3)/((x-1)^0.5)

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Answer

$2\frac{\frac{1}{7}\sqrt{x-1}x^{7}}{x^2}+\frac{12}{7}\cdot\frac{\frac{1}{5}\sqrt{x-1}x^{5}}{x^2}+\frac{32}{35}\sqrt{x-1}+\frac{48}{35}\cdot\frac{\frac{1}{3}\sqrt{x-1}x^{3}}{x^2}+C_0$

Step by step solution

Problem

$\int\frac{x^3}{\sqrt{x-1}}dx$
1

Solve the integral $\int\frac{x^3}{\sqrt{x-1}}$ by trigonometric substitution using the substitution

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

Substituting in the original integral, we get

$\int\frac{2\left(\sec\left(\theta\right)^{2}\right)^3\tan\left(\theta\right)\sec\left(\theta\right)^2}{\sqrt{\left(\sqrt{\sec\left(\theta\right)^{2}}\right)^{2}-1}}d\theta$
3

Applying the power of a power property

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

When multiplying exponents with same base we can add the exponents

$\int\frac{2\tan\left(\theta\right)\sec\left(\theta\right)^{8}}{\sqrt{\sec\left(\theta\right)^{2}-1}}d\theta$
5

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

$\int\frac{2\tan\left(\theta\right)\sec\left(\theta\right)^{8}}{\tan\left(\theta\right)}d\theta$
6

Simplifying the fraction by $\tan\left(\theta\right)$

$\int2\sec\left(\theta\right)^{8}d\theta$
7

Taking the constant out of the integral

$2\int\sec\left(\theta\right)^{8}d\theta$
8

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

$2\left(\frac{6}{7}\int\sec\left(\theta\right)^{6}d\theta+\frac{\sec\left(\theta\right)^{7}\sin\left(\theta\right)}{7}\right)$
9

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

$2\left(\frac{6}{7}\int\sec\left(\theta\right)^{6}d\theta+\frac{\frac{\sqrt{x-1}x^{7}}{x}}{7}\right)$
10

Simplifying the fraction

$2\left(\frac{6}{7}\int\sec\left(\theta\right)^{6}d\theta+\frac{\sqrt{x-1}x^{7}}{7x}\right)$
11

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

$2\left(\frac{6}{7}\int\sec\left(\theta\right)^{6}d\theta+\frac{\frac{1}{7}\cdot\frac{\sqrt{x-1}x^{7}}{x}}{x}\right)$
12

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

$2\left(\frac{6}{7}\left(\frac{4}{5}\int\sec\left(\theta\right)^{4}d\theta+\frac{\sec\left(\theta\right)^{5}\sin\left(\theta\right)}{5}\right)+\frac{\frac{1}{7}\cdot\frac{\sqrt{x-1}x^{7}}{x}}{x}\right)$
13

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

$2\left(\frac{6}{7}\left(\frac{4}{5}\int\sec\left(\theta\right)^{4}d\theta+\frac{\frac{\sqrt{x-1}x^{5}}{x}}{5}\right)+\frac{\frac{\frac{1}{7}\sqrt{x-1}x^{7}}{x}}{x}\right)$
14

Simplifying the fraction

$2\left(\frac{6}{7}\left(\frac{4}{5}\int\sec\left(\theta\right)^{4}d\theta+\frac{\sqrt{x-1}x^{5}}{5x}\right)+\frac{\frac{\frac{1}{7}\sqrt{x-1}x^{7}}{x}}{x}\right)$
15

Simplifying the fraction

$2\left(\frac{6}{7}\left(\frac{4}{5}\int\sec\left(\theta\right)^{4}d\theta+\frac{\sqrt{x-1}x^{5}}{5x}\right)+\frac{\frac{1}{7}\sqrt{x-1}x^{7}}{x\cdot x}\right)$
16

When multiplying exponents with same base you can add the exponents

$2\left(\frac{6}{7}\left(\frac{4}{5}\int\sec\left(\theta\right)^{4}d\theta+\frac{\sqrt{x-1}x^{5}}{5x}\right)+\frac{\frac{1}{7}\sqrt{x-1}x^{7}}{x^2}\right)$
17

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

$2\left(\frac{6}{7}\left(\frac{4}{5}\int\sec\left(\theta\right)^{4}d\theta+\frac{\frac{1}{5}\cdot\frac{\sqrt{x-1}x^{5}}{x}}{x}\right)+\frac{\frac{1}{7}\sqrt{x-1}x^{7}}{x^2}\right)$
18

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

$2\left(\frac{6}{7}\left(\frac{4}{5}\left(\frac{2}{3}\int\sec\left(\theta\right)^{2}d\theta+\frac{\sec\left(\theta\right)^{3}\sin\left(\theta\right)}{3}\right)+\frac{\frac{1}{5}\cdot\frac{\sqrt{x-1}x^{5}}{x}}{x}\right)+\frac{\frac{1}{7}\sqrt{x-1}x^{7}}{x^2}\right)$
19

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

$2\left(\frac{6}{7}\left(\frac{4}{5}\left(\frac{2}{3}\int\sec\left(\theta\right)^{2}d\theta+\frac{\frac{\sqrt{x-1}x^{3}}{x}}{3}\right)+\frac{\frac{\frac{1}{5}\sqrt{x-1}x^{5}}{x}}{x}\right)+\frac{\frac{1}{7}\sqrt{x-1}x^{7}}{x^2}\right)$
20

Simplifying the fraction

$2\left(\frac{6}{7}\left(\frac{4}{5}\left(\frac{2}{3}\int\sec\left(\theta\right)^{2}d\theta+\frac{\sqrt{x-1}x^{3}}{3x}\right)+\frac{\frac{\frac{1}{5}\sqrt{x-1}x^{5}}{x}}{x}\right)+\frac{\frac{1}{7}\sqrt{x-1}x^{7}}{x^2}\right)$
21

Simplifying the fraction

$2\left(\frac{6}{7}\left(\frac{4}{5}\left(\frac{2}{3}\int\sec\left(\theta\right)^{2}d\theta+\frac{\sqrt{x-1}x^{3}}{3x}\right)+\frac{\frac{1}{5}\sqrt{x-1}x^{5}}{x\cdot x}\right)+\frac{\frac{1}{7}\sqrt{x-1}x^{7}}{x^2}\right)$
22

When multiplying exponents with same base you can add the exponents

$2\left(\frac{6}{7}\left(\frac{4}{5}\left(\frac{2}{3}\int\sec\left(\theta\right)^{2}d\theta+\frac{\sqrt{x-1}x^{3}}{3x}\right)+\frac{\frac{1}{5}\sqrt{x-1}x^{5}}{x^2}\right)+\frac{\frac{1}{7}\sqrt{x-1}x^{7}}{x^2}\right)$
23

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

$2\left(\frac{6}{7}\left(\frac{4}{5}\left(\frac{2}{3}\int\sec\left(\theta\right)^{2}d\theta+\frac{\frac{1}{3}\cdot\frac{\sqrt{x-1}x^{3}}{x}}{x}\right)+\frac{\frac{1}{5}\sqrt{x-1}x^{5}}{x^2}\right)+\frac{\frac{1}{7}\sqrt{x-1}x^{7}}{x^2}\right)$
24

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

$2\left(\frac{6}{7}\left(\frac{4}{5}\left(\frac{2}{3}\tan\left(\theta\right)+\frac{\frac{1}{3}\cdot\frac{\sqrt{x-1}x^{3}}{x}}{x}\right)+\frac{\frac{1}{5}\sqrt{x-1}x^{5}}{x^2}\right)+\frac{\frac{1}{7}\sqrt{x-1}x^{7}}{x^2}\right)$
25

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

$2\left(\frac{6}{7}\left(\frac{4}{5}\left(\frac{2}{3}\sqrt{x-1}+\frac{\frac{\frac{1}{3}\sqrt{x-1}x^{3}}{x}}{x}\right)+\frac{\frac{1}{5}\sqrt{x-1}x^{5}}{x^2}\right)+\frac{\frac{1}{7}\sqrt{x-1}x^{7}}{x^2}\right)$
26

Simplifying the fraction

$2\left(\frac{6}{7}\left(\frac{4}{5}\left(\frac{2}{3}\sqrt{x-1}+\frac{\frac{1}{3}\sqrt{x-1}x^{3}}{x\cdot x}\right)+\frac{\frac{1}{5}\sqrt{x-1}x^{5}}{x^2}\right)+\frac{\frac{1}{7}\sqrt{x-1}x^{7}}{x^2}\right)$
27

When multiplying exponents with same base you can add the exponents

$2\left(\frac{6}{7}\left(\frac{4}{5}\left(\frac{2}{3}\sqrt{x-1}+\frac{\frac{1}{3}\sqrt{x-1}x^{3}}{x^2}\right)+\frac{\frac{1}{5}\sqrt{x-1}x^{5}}{x^2}\right)+\frac{\frac{1}{7}\sqrt{x-1}x^{7}}{x^2}\right)$
28

Multiply $\left(\frac{\frac{1}{3}\sqrt{x-1}x^{3}}{x^2}+\frac{2}{3}\sqrt{x-1}\right)$ by $\frac{48}{35}$

$2\frac{\frac{1}{7}\sqrt{x-1}x^{7}}{x^2}+\frac{12}{7}\cdot\frac{\frac{1}{5}\sqrt{x-1}x^{5}}{x^2}+\frac{32}{35}\sqrt{x-1}+\frac{48}{35}\cdot\frac{\frac{1}{3}\sqrt{x-1}x^{3}}{x^2}$
29

Add the constant of integration

$2\frac{\frac{1}{7}\sqrt{x-1}x^{7}}{x^2}+\frac{12}{7}\cdot\frac{\frac{1}{5}\sqrt{x-1}x^{5}}{x^2}+\frac{32}{35}\sqrt{x-1}+\frac{48}{35}\cdot\frac{\frac{1}{3}\sqrt{x-1}x^{3}}{x^2}+C_0$

Answer

$2\frac{\frac{1}{7}\sqrt{x-1}x^{7}}{x^2}+\frac{12}{7}\cdot\frac{\frac{1}{5}\sqrt{x-1}x^{5}}{x^2}+\frac{32}{35}\sqrt{x-1}+\frac{48}{35}\cdot\frac{\frac{1}{3}\sqrt{x-1}x^{3}}{x^2}+C_0$

Main topic:

Integration by trigonometric substitution

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