# Evaluate the indefinite integral. \int \cos 0\sin^{6}0 d0

Evaluate the indefinite integral.
$\int \mathrm{cos}0{\mathrm{sin}}^{6}0d0$
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Step 1
The given indefinite integral is $\int \mathrm{cos}0{\mathrm{sin}}^{6}0d0$
Solve by Substitution method
Substitute $\mathrm{sin}\theta =x$
Differentiate $\mathrm{sin}\theta =x$ with respect to x, we get
$\mathrm{cos}0d0=dx$
Step 2
Put $\mathrm{cos}\theta d\theta =dx$ in given integral
$\int {x}^{6}dx$
integrate $=\frac{{x}^{6+1}}{6+1}+c$   (applying $\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}$)
$=\frac{{x}^{7}}{7}+c$
Again substitute  $x=\mathrm{sin}\theta$
$\int \mathrm{cos}0{\mathrm{sin}}^{6}0d0=\frac{1}{7}{\mathrm{sin}}^{7}0+c$