`int_0^(pi/3) tan^5(x) sec^4(x) dx` Evaluate the integral

`tan^5(x)sec^4(x)=(sin^5(x))/(cos^9(x))=(1-cos^2(x))^2/(cos^9(x))*sin(x).`

Make a substitution `y=cos(x),` then `dy=-sin(x) dx` and the indefinite integral is equal to

`-int (1-y^2)^2/y^9 dy=-int(y^(-9)-2y^(-7)+y^(-5)) dy=`

`=1/8 y^(-8)-1/3 y^(-6)+1/4 y^(-4)+C.`

And the definite integral is equal to

`(1/8 y^(-8)-1/3 y^(-6)+1/4 y^(-4))|_(y=cos(0)=1)^(y=cos(pi/3)=1/2)=`

`=1/8(2^8-1)-1/3(2^6-1)+1/4(2^4-1)=255/8-21+30/8=117/8.`