Let I=\int\limits^4_(-2) {x^4cos(x^5+2)} \, dx. Use the substitution u=x^5+2 to convert I into an equivalent integral in the variable u.

Question

Let
`I=\int\limits^4_(-2) {x^4cos(x^5+2)} \, dx`. Use the substitution
`u=x^5+2` to convert
`I` into an equivalent integral in the variable
`u`.

Answer 1

If
`u=x^5+2`, then
`du=5x^4\,dx`, so the integral transforms to

`\displaystyle \int_(x=-2)^(x=4) x^4 \cos(x^5+2) \, dx = \frac15 \int_(u=-30)^(u=1026) \cos(u) \, du`

since

`x = -2 \implies u = (-2)^5 + 2 = -32 + 2 = -30`

`x=4 \implies u = 4^5 + 2 = 1024 + 2 = 1026`