indefinite integral(e^(3x)sin(4x)dx) integrate by parts: u=e^(3x) dv=sin(4x)dx du=3e^(3x)dx v=(-1/4)cos(4x) indefinite integral = (-1/4)e^(3x)cos(4x) - indefinite integral((-3/4)e^(3x)cos(4x)dx) Rearrange fractions: indefinite integral = (-1/4)e^(3x)cos(4x) + (3/4) * indefinite integral(e^(3x)cos(4x)dx) integrate by parts: u=e^(3x) dv=cos(4x)dx du=3e^(3x)dx v=(1/4)sin(4x) indefinite integral = (-1/4)e^(3x)cos(4x) + (3/4)((1/4)e^3xsin(4x)-((3/4) * indefinite integral(e^(3x)sin(4x)dx)) Rearrange and multiply out fractions: indefinite integral = (-1/4)e^(3x)cos(4x) + (3/16)e^3xsin(4x)-((9/16) * indefinite integral(e^(3x)sin(4x)dx) Consolidate similar indefinite integrals: (25/16) * indefinite integral = (-1/4)e^(3x)cos(4x) + (3/16)e^3xsin(4x) Isolate the indefinite integral: indefinite integral = (-4/25)e^(3x)cos(4x)+(3/25)e^(3x)sin(4x)+c
