1. ## integration help

Need help with these examples:

a) $\int \frac{\sqrt{1+2lnx}}{x}\, dx$

b) $\int x^5\sqrt{1-x^3}\, dx$

2. $

\int x^5\sqrt{1-x^3}\, dx
$

Let $u=1-x^3 \Rightarrow \frac{du}{dx}-3x^2$

$\int x^5 \sqrt{u} \frac{du}{-3x^2} \Rightarrow -\frac{1}{3}\int x^3 \sqrt{u} ~du$

$\Rightarrow -\frac{1}{3} \int (1-u)\sqrt{u}~du$

$=-\frac{1}{3} \int \sqrt{u}-u^{\frac{3}{2}}~du$

$=-\frac{1}{3} \left( \frac{2}{3}u^{\frac{3}{2}}-\frac{2}{5} u^{\frac{5}{2}} \right)+C$

$=-\frac{2}{3} \left( \frac{1}{3}u^{\frac{3}{2}}-\frac{1}{5} u^{\frac{5}{2}} \right)+C$

3. $

\int \frac{\sqrt{1+2lnx}}{x}\, dx
$

Let $u=1+2ln \ x \Rightarrow \frac{du}{dx}=\frac{2}{x}$

$\int \frac{\sqrt{u}}{x}~\frac{x~du}{2}$

$\frac{1}{2}\int \sqrt{u}~du$

$\frac{1}{2}.\frac{2}{3} u^{\frac{3}{2}}+C$

$\frac{1}{3} u^{\frac{3}{2}}+C$

4. Originally Posted by Bernice
Need help with these examples:

a) $\int \frac{\sqrt{1+2lnx}}{x}\, dx$

b) $\int x^5\sqrt{1-x^3}\, dx$
a) Substitute $u = 1 + 2 \ln x$.

b) Substitute $u = 1 - x^3$.

5. a more direct substitution is to replace those $u$ by $u^2.$