Question

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

Answer 1

Zero

Explanation:

`int_0^(4sqrtpi)xcos(5x^2)dx`

Use `x^2` as your direct variable instead of `x`:
`d(x^2)=2xdx`

Hence
`int_0^(4sqrtpi)xcos(5x^2)1/(2x)d(x^2)`
`1/2int_0^(4sqrtpi)cos(5x^2)d(x^2)`
`1/2sin(5x^2)` from `0 to 4sqrtpi`

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