# Find the trigonometric integral \int \sin x * \cos^{4} x

Find the trigonometric integral $\int \mathrm{sin}x\cdot {\mathrm{cos}}^{4}xdx$
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Step 1
Consider the integrals:
$\int \mathrm{sin}x\cdot {\mathrm{cos}}^{4}xdx$...(1)
Let, $t=\mathrm{cos}x$
Differentiating   w.r.t.t
$dt=-\mathrm{sin}xdx⇒\mathrm{sin}xdx=-dt$
Step 2
Substitute all value in equation (1) then,
$\mathrm{sin}x\cdot {\mathrm{cos}}^{4}xdx=\int {t}^{4}\left(-dt\right)$
$=-\int {t}^{4}dt$
$=-\frac{{t}^{5}}{5}+C$
Back substitute $t=\mathrm{cos}x$
$=-\frac{1}{5}{\mathrm{cos}}^{5}x+C$