Question

integration of {2 + sinx}^{3} cosxdx

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Answer 1

Step-by-step explanation:

We have,

$\int(2 + \sin(x))^{3} \cos(x) dx$

$= \int(8 \cos(x) + \sin^{3} (x) \cos(x) + 12 \sin(x) \cos(x) + 6 \sin^{2} (x) \cos(x) )dx$

$= \int8 \cos(x) dx + \int\sin^{3} (x) \cos(x) dx + \int 12 \sin(x) \cos(x) dx + \int6 \sin^{2} (x) \cos(x) dx$

Let sin(x) = t

=> cos(x) dx = dt

$= 8\int\cos(x) dx + \int \: t^{3} dt +12 \int t dt + 6\int t^{2} dt$

$= 8 \sin(x) + \frac{ {t}^{4} }{4} +12 . \frac{ {t}^{2} }{2} + 6. \frac{ t^{3}}{3}$

$= 8 \sin(x) + \frac{ \sin^{4} (x) }{4} + 6 \sin^{2} (x) + 2 \sin^ {3} (x)$