Examples from Reduction formulae

$\int x^3 e^x \, dx$

The recurrence with $n=3$, then $n=2$, then $n=1$ and $I_0=\int e^x \, dx$, gives

$\int_0^{\pi/2}\sin^5 x \, dx$

$\int_0^{\pi/2}\cos^7 x \, dx$

Secant and tangent recurrences on $\bigl[0,\frac{\pi}{4}\bigr]$

The base value $I_2=\int_0^{\pi/4}\sec^2 x \, dx=\bigl[\tan x\bigr]_0^{\pi/4}=1$ gives

The base value $J_2=\int_0^{\pi/4}\tan^2 x \, dx=\int_0^{\pi/4}(\sec^2 x-1)\, dx=\bigl[\tan x-x\bigr]_0^{\pi/4}=1-\frac{\pi}{4}$ gives

For mixed powers, $K_1=\int_0^{\pi/4}\tan\theta\sec\theta \, d\theta=\bigl[\sec\theta\bigr]_0^{\pi/4}=\sqrt{2}-1$, so

$\int_0^{\pi/2}\cos^5 x\sin^4 x \, dx$

This is the definite version of the odd-power evaluation of $\int\cos^5 x\sin^4 x \, dx$. Reduce the cosine power twice, then the sine power twice and use $I_{1,0}=1$: