Evaluate the following integrals:

Question:

Evaluate $\int \cos ^{5} x d x$.

Solution:

$y=\int\left(1-\sin ^{2} x\right)^{2} \cos x d x$

Let, $\sin x=t$

Differentiating both side with respect to $x$

$\frac{d t}{d x}=\cos x \Rightarrow d t=\cos x d x$

$y=\int\left(1-t^{2}\right)^{2} d t$

$y=\int 1+t^{4}-2 t^{2} d t$

Using formula $\int t^{n} d t=\frac{t^{n+1}}{n+1}$ and $\int c d t=c t$

$y=\left(t+\frac{t^{5}}{5}-2 \frac{t^{3}}{3}\right)+c$

Again, put $t=\sin x$

$y=\left(\sin x+\frac{\sin ^{5} x}{5}-2 \frac{\sin ^{3} x}{3}\right)+c$

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