The formula for integrating -cos^2x sinx is ∫-cos^2x sinx dx = -∫cos^2x d(cosx).
Step 1: Use the identity cos2x = 1 - 2sin^2x to rewrite the integral as ∫-(1 - 2sin^2x)sinx dx.Step 2: Expand the expression to get ∫-sinx + 2sin^3x dx.Step 3: Use the power rule for integration to get -∫sinx dx + 2∫sin^3x dx.Step 4: Solve the first integral using the substitution u = sinx, and the second integral using the reduction formula for sin^nx.Step 5: Substitute back for u and simplify to get the final answer.
The result of integrating -cos^2x sinx is -sinx + (1/4)sin^4x + C.
Yes, the trick to integrating -cos^2x sinx is to use the identity cos2x = 1 - 2sin^2x to rewrite the integral and then use the power rule for integration and reduction formula for sin^nx to solve it.
Example: ∫-cos^2x sinx dx= -∫cos^2x d(cosx)= -∫(1 - 2sin^2x) d(cosx)= -∫d(cosx) + 2∫sin^2x d(cosx)= -cosx + 2∫sin^2x d(cosx)= -cosx + 2(1/2)sin^2x cosx + C= -cosx + (1/2)sin^4x + C.