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.

asked Jul 14, 2023 17.4k views

5 votes

Let
$I=\int\limits^4_(-2) {x^4cos(x^5+2)} \, dx$ . Use the substitution
u=x^5+2

to convert

into an equivalent integral in the variable

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6 votes

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

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

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

Arniotaki answered Jul 19, 2023

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