Question

For ∫sin⁷ (theta) cos (theta) dtheta, use a suitable change of variables to determine the indefinite integral.

Answer 1

The indefinite integral of sin⁷(θ) cos(θ) dθ is (1/8)sin⁸(θ) + C, where C is the constant of integration.

For the integral ∫sin⁷(θ) cos(θ) dθ, you can use a trigonometric identity to simplify it.

The substitution you can make here is u = sin(θ), and then du = cos(θ) dθ.

Start by expressing cos(θ) dθ in terms of du:

du = cos(θ) dθ

Now, rewrite the integral using the substitution u = sin(θ) and du = cos(θ) dθ:

∫sin⁷(θ) cos(θ) dθ = ∫u^7 du

Integrating ∫u^7 du:

∫u^7 du = (1/8)u^8 + C

Finally, replace u with sin(θ):

(1/8)sin⁸(θ) + C

So, the indefinite integral of sin⁷(θ) cos(θ) dθ is (1/8)sin⁸(θ) + C, where C is the constant of integration.