$\int sin^5{x}dx=?$ | What is integral of sinx^5

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What is integral of ∫sinx^5dx?

integrate of sinx^5

Evaluate $\int sin^5{x}dx$.

Rewrite the function:

$\int sin^5{x}.dx=\int sinx.sin^4x.dx=\int sinx(sin^2x)^2.dx=\int sinx(1-cos^2x)^2.dx$

Now use u=cosx, du=-sinxdx:

$\int sinx(1-cos^2x)^2.dx=\int -(1-u^2)^2du=\int -(1-2u^2+u^4)du$

=$-u+\frac{2}{3}u^3-\frac{1}{5}u^5+C$

=$-cosx+\frac{2}{3}cos^3x-\frac{1}{5}cos^5x+C$

Answer: 

$\int sin^5{x}.dx=-cosx+\frac{2}{3}cos^3x-\frac{1}{5}cos^5x+C$