How do you find the definite integral of $cos^4 x sin x dx$ in the interval $[0, pi/3]$ ?

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Answer 1 · 2016-10-06T08:22:45

$31/160$

Knowing that

$d/(dx)cos^n x = -ncos^(n-1)x sin x$ and making $n = 5$ we have

$-5cos^4 xsin x = d/(dx)cos^5x$ so

$cos^4 x sin x = -1/5d/(dx)cos^5 x$ and consequently

$int_0^(pi/3) cos^4x sin x dx = -1/5cos^5(pi/3) +1/5cos^5(0)=1/5(1-cos^5(pi/3))=31/160$

How do you find the definite integral of cos^4 x sin x dxcos^4 x sin x dx in the interval [0, pi/3][0, pi/3]?