# \int

• Jan 17th 2013, 09:43 AM
marijakopljar
\int
Hello!
Please, any good soul to help me with:
$\intx^2sin2xdx$???
The result is :
$\frac{1}{4}(1-2x^2)cos2x+\frac{1}{2}xsin2x+C$

Many thanks!!!
• Jan 17th 2013, 10:42 AM
HallsofIvy
Re: \int
Quote:

Originally Posted by marijakopljar
Hello!
Please, any good soul to help me with:
$\int x^2sin2xdx$???

You left out the space between 'int' and 'x^2'. I added it.

Use "integration by parts": $\int u dv= uv- \int v du$ with $u= x^2$, $dv= sin(2x)dx$ so that $du= 2xdx$, $v= -\frac{1}{2}cos(2x)$. Doing that, you will still have an integral involving $xcos(2x)$. So use integration by parts again, this time letting $u= x$, $dv= cos(2x)dx$.

Generally speaking, any time you have an integral of $x^n f(x)dx$, you can consider repeated integration by parts, repeatedly taking the power of x as "u" so that when you differentiate to get du, you have one less power of x. Repeat until you have reduced the power of x to 0.

Quote:

The result is :
$\frac{1}{4}(1-2x^2)cos2x+\frac{1}{2}xsin2x+C$

Many thanks!!!