Question

Evaluate the integral by making the given substitution. ∫ cos¹⁷θsinθ dθ, u = cos θ

Answer 1

To evaluate the integral ∫ cos⁹¹θsinθ dθ by making the given substitution u = cos θ, we rewrite the integral using a trigonometric identity and substitute the variable. We then integrate and substitute back to obtain the final answer.

Step-by-step explanation:

To evaluate the integral ∫ cos⁹¹θsinθ dθ by making the given substitution u = cos θ, we can use the trigonometric identity sin²θ = 1 - cos²θ. Let's rewrite the integral using this identity: ∫ cos⁹¹θsinθ dθ = ∫ cos⁹¹θ(1-cos²θ) dθ. Now, substitute u = cos θ and dθ = -du / sinθ to get: ∫ (u-1)du. Integrating this expression gives us: ½u⁺¹ - u + C, where C is the constant of integration. Finally, substitute back u = cos θ to get the final answer: ½cos⁺¹θ - cos θ + C.