Integration by Parts

Integration By Parts (The Worst Technique)

Differentiate a function that is the product of 2 simpler functions. It is an Iterative Process $\int u^{\prime }vdx=uv-\int u{v}^{\prime }dx$

1. Assign $\int u^{\prime }$ and $\int v$
2. Determine $u$ and $v^{\prime }$
3. Check $\int u^{\prime }vdx=uv-\int u{v}^{\prime }dx$
4. There is one $+c$ at the very end, so you dont need to have $+c$ midway.

Wrong Example

Determine $\int x\sin xdx$

1. $u^{\prime }=x$, and then naturally, $v=\sin x$
2. Therefore, $u=\frac{x^2}{2}+c$ as the antiderivative of $u^{\prime }$. $v^{\prime }=\cos x$
3. Check $uv-\int u{v}^{\prime }dx=\frac{x^2}{2}\sin x-\int \frac{x^2}{2}dx$.
   1. It is too hard to integrate $\int \frac{x^2}{2}dx$, therefore, this was a bad decision of $u^{\prime }$ and $v$

Right Example

Determine $\int x\sin xdx$

1. $u^{\prime }=\sin x$, $v=x$
2. $u=-\cos x$, $v^{\prime }=1$
3. $uv-\int u{v}^{\prime }dx=-x\cos x-\int (-\cos x)dx$
4. Now solve, $\int x\sin xdx=-x\cos x+\sin x+c$