∫ sin ( 5.75 x ) ^8.25 cos ( 5.75 x )^ 5 d x

Question

solve the inegral using trignometric ticheque of integration.

Answer 1

Let u=sin(5.75x), du=5.75cos(5.75x)dx, and the integral becomes:

(1/5.75)∫(sin(5.75x))^8.25cos⁴(5.75x)du.

We need the integrand in terms of u. cos⁴(5.75x)=(1-sin²(5.75x))²=1-2sin²(5.75x)+sin⁴(5.75x)=1-2u²+u⁴.

Now we have (4/23)∫u^8.25(1-2u²+u⁴)du=(4/23)∫(u^8.25-2u^10.25+u^12.25)du.

Integral is (4/23)(u^9.25/9.25-2u^11.25/11.25+u^13.25/13.25)+C.

Substitute u=sin(5.75x) to get the answer in terms of x.