1. ## integration by parts

can someone show me step by step in solving these problems?

$\displaystyle \int{x^2 sin4x^3} {dx}$

and

$\displaystyle \int{sin^3 2x cos^2 2x} {dx}$

thx in advance of ur help

2. Originally Posted by kokolily
can someone show me step by step in solving these problems?

$\displaystyle \int{x^2 sin4x^3} {dx}$

and

$\displaystyle \int{sin^3 2x cos^2 2x} {dx}$

thx in advance of ur help
Neither of these are integration by part, but are u substitions.

For the first one use the u sub $\displaystyle u=4x^3 \implies du=12x^2dx$ to get

$\displaystyle \int{x^2 sin4x^3} {dx} =\frac{1}{12}\int \sin(u)du$

For the 2nd one you need a trig identity...

$\displaystyle \int{\sin^3 2x \cos^2 2x} {dx} =\int \sin(2x)[1-\cos^2(2x)]\cos^2(2x)dx=$

$\displaystyle \int \sin(2x)\cos^2(2x)dx-\int \sin(2x)\cos^4(2x)dx$

Now just let $\displaystyle u=\cos(2x)$ and you are off to the races.

3. thanks for ur help

how do u know that it is not int by parts but exactly int by substitution?