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lunes, 9 de mayo de 2011

Integral by Substitution Example #4

$\int sinx*cosx\;dx$

$u= sinx$

$du= cosx \; dx$

$ \int u\;du= \frac{1}{2} u^2+c $

$ \frac{1}{2} sin^2x+c $

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