an integral from 0 to pi/2 the function is |5sin(x)-5cos(2x)|dx

This integral is a bit complicated because of the absolute value. The argument of the abs function is negative (or zero) in the range 0 to π/6 and positive (or zero) in the range π/6 to π/2. We must break up the range of integration into two parts:
0 to π/6 and then π/6 to π/2.

The integrand in the first part is 5(cos(2x) - sin(x) ) and the integrand in the second part is

5 ( sin(x) - cos(2x) ). We must evaluate the two definite integrals and then add to get the final answer.

The antiderivative of sin(x) is - cos(x) and the antiderivative of cos(2x) is sin(2x) /2.

When all the pieces are put together, the final result is 5 ( (3/2) sqrt(3) -1 ) ~ 7.99