1. ## integration help

Need help with these examples:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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