Integral of (x+2)(x-2)^0.5

\int\left(x+2\right)\sqrt{x-2}dx

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Answer

$\frac{2}{5}\sqrt{\left(x-2\right)^{5}}+\frac{28}{9}\sqrt{\left(x-2\right)^{3}}+C_0$

Step by step solution

Problem

$\int\left(x+2\right)\sqrt{x-2}dx$
1

Multiplying polynomials $\sqrt{x-2}$ and $x+2$

$\int\left(2\sqrt{x-2}+x\sqrt{x-2}\right)dx$
2

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

$\int2\sqrt{x-2}dx+\int x\sqrt{x-2}dx$
3

Taking the constant out of the integral

$2\int\sqrt{x-2}dx+\int x\sqrt{x-2}dx$
4

Apply the formula: $\int\left(a+x\right)^ndx$$=\frac{\left(a+x\right)^{\left(1+n\right)}}{1+n}$, where $a=-2$ and $n=\frac{1}{2}$

$2\frac{\sqrt{\left(x-2\right)^{3}}}{\frac{3}{2}}+\int x\sqrt{x-2}dx$
5

Simplify the fraction

$\frac{4}{3}\sqrt{\left(x-2\right)^{3}}+\int x\sqrt{x-2}dx$
6

Solve the integral $\int x\sqrt{x-2}dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=x-2 \\ du=dx\end{matrix}$
7

Rewriting $x$ in terms of $u$

$x=2+u$
8

Substituting $u$, $dx$ and $x$ in the integral

$\frac{4}{3}\sqrt{\left(x-2\right)^{3}}+\int\left(2+u\right)\sqrt{u}du$
9

Multiplying polynomials $\sqrt{u}$ and $u+2$

$\frac{4}{3}\sqrt{\left(x-2\right)^{3}}+\int\left(2\sqrt{u}+\sqrt{u^{3}}\right)du$
10

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

$\frac{4}{3}\sqrt{\left(x-2\right)^{3}}+\int2\sqrt{u}du+\int\sqrt{u^{3}}du$
11

Taking the constant out of the integral

$\frac{4}{3}\sqrt{\left(x-2\right)^{3}}+2\int\sqrt{u}du+\int\sqrt{u^{3}}du$
12

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\frac{4}{3}\sqrt{\left(x-2\right)^{3}}+2\cdot \frac{2}{3}\sqrt{u^{3}}+\int\sqrt{u^{3}}du$
13

Substitute $u$ back for it's value, $x-2$

$\frac{4}{3}\sqrt{\left(x-2\right)^{3}}+\frac{4}{3}\sqrt{\left(x-2\right)^{3}}+\int\sqrt{u^{3}}du$
14

Adding $\frac{4}{3}\frac{4}{3}\sqrt{\left(x-2\right)^{3}}$ and $\frac{4}{3}\sqrt{\left(x-2\right)^{3}}$

$\int\sqrt{u^{3}}du+\frac{7}{3}\cdot \frac{4}{3}\sqrt{\left(x-2\right)^{3}}$
15

Multiply $\frac{4}{3}$ times $\frac{7}{3}$

$\int\sqrt{u^{3}}du+\frac{28}{9}\sqrt{\left(x-2\right)^{3}}$
16

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\frac{2}{5}\sqrt{u^{5}}+\frac{28}{9}\sqrt{\left(x-2\right)^{3}}$
17

Substitute $u$ back for it's value, $x-2$

$\frac{2}{5}\sqrt{\left(x-2\right)^{5}}+\frac{28}{9}\sqrt{\left(x-2\right)^{3}}$
18

Add the constant of integration

$\frac{2}{5}\sqrt{\left(x-2\right)^{5}}+\frac{28}{9}\sqrt{\left(x-2\right)^{3}}+C_0$

Answer

$\frac{2}{5}\sqrt{\left(x-2\right)^{5}}+\frac{28}{9}\sqrt{\left(x-2\right)^{3}}+C_0$

Problem Analysis

Main topic:

Integration by substitution

Time to solve it:

0.33 seconds

Views:

108