## lunes, 9 de mayo de 2011

### Integral by Substitution Example #3

Calculate
$\int_0^{\sqrt{\pi}}x\cos (x^2)\;dx$

SOLUTION
Let
\begin{align*} u &= x^2 \\ du &= 2\,dx \end{align*}

To find the limits of integration we note that when,

$x=0 \rightarrow u = 0^2 = 0$

and when

$x=\sqrt{\pi} \rightarrow u = \sqrt{\pi}^{\;2} = \pi$

Therefore
\begin{align*} \int_0^{\sqrt{\pi}}x\cos (x^2)\;dx&=\frac{1}{2}\int_0^{\pi}\cos (u)\;du \\ &= \left. \frac{1}{2} \sin(u) \right |_0^\pi\\ &= \frac{1}{2} \left ( \sin(\pi)-\sin (0) \right ) \\ &= 0 \end{align*}

#### 6 comentarios:

1. It was only a year ago that I took calculus, yet I don't remember any of this. Thanks for reminding me how it's done!

2. I have to write an exam about this stuff tomorrow, yet I have no clue what this is supposed to mean. I think I have to study through the night :/

3. I loved this class!

4. Thanks for helping me out bro.

5. Fuckkk, math is my worst subject... LOL