Heres a specific type of integral that can be evaluated much easily using the gamma function than integrating by parts


₀^pi/2 cos^mx sinⁿx dx = [g{(m+1)/2} * g{(n+1)/2}]/2*g{(m+n+2)/2} where m,n belong to {0,1,2,3....}

where g is the gamma function

g(n)=(n-1)!       if nN

g(n/2)= (n/2-1)*((n-2)/2-1).......1/2  *  sqrt(pi)

This seems to be too complicated but it is simple to use
Here is an example

Evaluate I = ₀^pi/2 cos⁶x sin⁴x dx

here m=6 n=4
=> I = g{(6+1)/2} * g{(4+1)/2}  /  2*g{(6+4+2)/2}
=> I = g(7/2)*g((5/2) / 2*g(6)

now using the above mentioned definitions

g(7/2)=5/2 * 3/2 * 1/2 * sqrt(pi)
g(5/2)=3/2*1/2*sqrt(pi)
g(6)=5!=120

=> I=5*3*3*pi/2*2*2*2*2*120
=> I=45pi/3840
=> I=3pi/256

Heres another example,

Evaluate I = ₀^pi/2 cos¹¹x dx
=>I = ₀^pi/2 cos¹¹x sin⁰x dx
again using the gamma function
=>I= {g(6)*g(1/2)}/2g(13/2)
=>I=5!*sqrt(pi)/{11/2*9/2*7/2*5/2*3/2*1/2*sqrt(pi)
=>I=120*2⁶/*2{11*9*7*5*3}
=>I=7680/10395*2
=>I=768/2079

in cases when the limits of the integral are not 0 to pi/2 the limits are to be manipulated to become 0 to pi/2 and then again this method is applicable.

i cannot explain what the gamma function actually means at my level as this is something to be studied after class XII. Though it can be used in the integral of the given type.

Hope it is useful!