Question

Integrate the given function w.r.t. respective variable : cos³ x

Answer 1

this is the answer of the integrated function of cos^3x

Answer 2

Answer:

$\int\:cos^{3}x\:dx=sin\:x-\frac{sin^{3}x }{3} +C$

Step-by-step explanation:

We know that before integrating any function ,we must convert it into integrable form.

$\int\:cos^{3}x\:dx\\ \\ =\int\:cos^{2}x.cos\:x\:dx\\ \\=\int\:(1-sin^{2}x).cos\:x\:dx$

now by analysis we can see that differentiation of  sin x is cos x,so we can substitute

$sin\:x =t\\ \\ so\\ \\cos\:x\:dx=dt\\ \\substitute\:\:\\ \\ \int\:(1-t^{2} )dt\\ \\ =\int\:1.dt-\int\:t^{2} \:dt\\ \\ =t-\frac{t^{3} }{3} +C\\ \\undo\:\:substitution\\ \\ \\ \int\:cos^{3}x\:dx=sin\:x-\frac{sin^{3}x }{3} +C$