# How do you find the definite integral for: xcos(5x^2) dx for the intervals [0,4sqrtpi]?

Apr 1, 2016

Zero

#### Explanation:

${\int}_{0}^{4 \sqrt{\pi}} x \cos \left(5 {x}^{2}\right) \mathrm{dx}$

Use ${x}^{2}$ as your direct variable instead of $x$:
$d \left({x}^{2}\right) = 2 x \mathrm{dx}$

Hence
${\int}_{0}^{4 \sqrt{\pi}} x \cos \left(5 {x}^{2}\right) \frac{1}{2 x} d \left({x}^{2}\right)$
$\frac{1}{2} {\int}_{0}^{4 \sqrt{\pi}} \cos \left(5 {x}^{2}\right) d \left({x}^{2}\right)$
$\frac{1}{2} \sin \left(5 {x}^{2}\right)$ from $0 \to 4 \sqrt{\pi}$

You can immediately see that the sine term will give zero value for both limits.