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Step-by-step Solution

Integrate $x\left(1+x\right)^{\left(\frac{1}{2}\right)}$ with respect to x

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Answer

$\frac{2}{5}\sqrt{\left(1+x\right)^{5}}-\frac{2}{3}\sqrt{\left(1+x\right)^{3}}+C_0$

Step-by-step explanation

Problem to solve:

$\int\left(X\left(1+X\right)^{\left(1/2\right)}\right)dx$
1

Solve the integral $\int x\sqrt{1+x}dx$ by trigonometric substitution using the substitution

$\begin{matrix}x=\left(1\tan\left(\theta\right)\right)^{2} \\ dx=2\sec\left(\theta\right)^2\tan\left(\theta\right)d\theta\end{matrix}$
2

Substituting in the original integral, we get

$\int2\left(1\tan\left(\theta\right)\right)^{2}\sqrt{1+\left(\sqrt{\left(1\tan\left(\theta\right)\right)^{2}}\right)^{2}}\sec\left(\theta\right)^2\tan\left(\theta\right)d\theta$

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Answer

$\frac{2}{5}\sqrt{\left(1+x\right)^{5}}-\frac{2}{3}\sqrt{\left(1+x\right)^{3}}+C_0$

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