Question

Find the indefinite integral of cos⁵(t)sin(t) dt. (Use c for the constant of integration.)

Answer 1

To find the indefinite integral of cos⁵(t)sin(t) dt, we can use integration by parts.

Step-by-step explanation:

To find the indefinite integral of cos⁵(t)sin(t) dt, we can use integration by parts. Let u = cos⁵(t) and dv = sin(t) dt. Taking the derivative of u and integrating dv, we have du = -5cos⁴(t)sin(t) dt and v = -cos(t) dt. Applying the integration by parts formula, we get:

∫cos⁵(t)sin(t) dt = -cos⁵(t)cos(t) + ∫5cos⁴(t)cos(t) dt

Simplifying the integral, we get:

∫cos⁵(t)sin(t) dt = -cos⁶(t)/6 + 5/6∫cos⁴(t)cos(t) dt

We can continue applying integration by parts until we reach an integral that we can easily solve.