What is the integral of $int cos^3 (x)dx$ from 0 to pi/2?

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Answer 1 · 2017-02-08T19:49:07

$int_0^(pi/2) cos^3xdx = 2/3$

Use the idendity $cos^2x+sin^2x = 1$ :

$int_0^(pi/2) cos^3xdx = int_0^(pi/2) cosx cos^2xdx = int_0^(pi/2) cosx (1-sin^2x) xdx$

substitute:

$t=sinx$
$dt = cosx dx$
$x in (0,pi/2) => t in (0,1)$

so:

$int_0^(pi/2) cos^3xdx = int_0^1 (1-t^2)dt = [t-t^3/3]_0^1 = 2/3$

What is the integral of int cos^3 (x)dxint cos^3 (x)dx from 0 to pi/2?