Evaluate \int \sin^{5}xdx.

Step 1
In some cases integrals can be simplified by use of a substitution. This usually results in simplifying the integral in terms of known standard integrals.
For the given integral the integrand can be written as ${\displaystyle \sin \left(x\right){\sin }^4\left(x\right)=\sin \left(x\right){(1-{\cos }^2\left(x\right))}^2}$.
Then the substitution ${\displaystyle u=\cos \left(x\right)}$ will simplify the integrand expression.
Step 2
The given indefinite integral to be evaluated is ${\displaystyle \int {\sin }^5\left(x\right)dx}$. This can be rewritten as ${\displaystyle \int \sin \left(x\right){(1-{\cos }^2\left(x\right))}^{2}dx}$.Use the substitution ${\displaystyle u=\cos \left(x\right)}$. Differentiating this gives ${\displaystyle -du=\sin \left(x\right)dx}$.Substitute this in the integral and evaluate the integral.
${\displaystyle \int {\sin }^5\left(x\right)dx=\int \sin \left(x\right){(1-{\cos }^2\left(x\right))}^{2}dx}$
${\displaystyle =-\int {(1-u^2)}^{2}du}$
${\displaystyle =-\int (1+u^4-2u^2)du}$
${\displaystyle =-u-\frac{u^5}{5}+\frac{2u^3}{3}+C}$
${\displaystyle =-\cos \left(x\right)-\frac{{\cos }^5\left(x\right)}{5}+\frac{2{\cos }^3\left(x\right)}{3}+C}$