Evaluate the indefinite integral x cos8x dxSolution integral

Evaluate the indefinite integral.

x cos(8x) dx

Solution

integral x cos(8 x) dx For the integrand x cos(8 x), integrate by parts, integral f dg = f g- integral g df, where f = x, dg = cos(8 x) dx, df = dx, g = 1/8 sin(8 x): = 1/8 x sin(8 x)-1/8 integral sin(8 x) dx For the integrand sin(8 x), substitute u = 8 x and du = 8 dx: = 1/8 x sin(8 x)-1/64 integral sin(u) du The integral of sin(u) is -cos(u): = (cos(u))/64+1/8 x sin(8 x)+constant Substitute back for u = 8 x: = 1/8 x sin(8 x)+1/64 cos(8 x)+constant Which is equal to: = 1/64 (8 x sin(8 x)+cos(8 x))+constant