# How to integrate int12x[sqrt(x+1)]dx using integration by parts ?

##### 1 Answer
Apr 30, 2018

The answer is $= 8 x {\left(1 + x\right)}^{\frac{3}{2}} - \frac{16}{5} {\left(1 + x\right)}^{\frac{5}{2}} + C$

#### Explanation:

We need

We solve this integral by integration by parts

$\int u v ' \mathrm{dx} = u v - \int u ' v$

Here,

$u = \left(12 x\right)$, $\implies$, $u ' = 12$

$v ' = \sqrt{x + 1}$, $\implies$, $v = \frac{2}{3} {\left(x + 1\right)}^{\frac{3}{2}}$

Therefore,

$\int 12 x \sqrt{x + 1} \mathrm{dx} = 12 x \cdot \frac{2}{3} {\left(1 + x\right)}^{\frac{3}{2}} - \int 12 \cdot \frac{2}{3} {\left(1 + x\right)}^{\frac{3}{2}} \mathrm{dx}$

$= 8 x {\left(1 + x\right)}^{\frac{3}{2}} - 8 \cdot \frac{2}{5} \cdot {\left(1 + x\right)}^{\frac{5}{2}} + C$

$= 8 x {\left(1 + x\right)}^{\frac{3}{2}} - \frac{16}{5} {\left(1 + x\right)}^{\frac{5}{2}} + C$