Evaluate the integrals in Exercises 33–52.
∫ 8 cot⁴(t) d...

Question

Evaluate the integrals in Exercises 33–52.

∫ 8 cot⁴(t) dt

Accepted Answer

Welcome back everyone. Evaluate the integral co sequence to the power of 4 of x d x. For this problem, let's remember one of the trigonometric identities. It is co sequent squared of x equals. Co tangent squared of x + 1. We're going to use it in this problem. In order to use it, let's rewrite co sequent to the power of 4 of x as co sequin squared of x multiplied by co sequin squared of x. So now this is going to be equal to, let's leave one co sequin squared, so that'd be co sequin squared x, and for the other one we will rewrite it as cotangent squared of x + 1. And now if we distribute that would be co sequin square x cotangent squared x plus co sequin square x. So now we can rewrite our integral as the integral of co sequin square x cotangent squared x dx, plus another integral of cosequin squared xdx. Now for the first integral, we're going to use u substitution. Let's suppose that u is equal to cotangent x. The derivative of u with respect to x is the derivative of cotangent x, which is negative co sequin squared x. And if we multiply both sides by negative DX, we're going to get DU is equal to cos 2 X. D X This is exactly what we have within the integral co sequin squared x dx. So the first integral can be rewritten as u 2 followed by negative duu. So we can write DU and we can factor out the negative sign. And we're going to integrate using the power rule that's you raise the power of 2 + 1, so that's u cubed divided by 3. Plus a constant of integration c 1. And now since u is cotangent x, we get negative. 1/3. Cotangent cub x + C1. The second integral is an elementary integral, so we can immediately combine them. We're going to get 1/3 cotangent cubed x from the first one. Let's not worry about the constant c1 for now. The second integral is the integral of co sequin squared x dx, and its value is negative cotangent x, right? This is an elementary integral, and then we will have a combination of integration constants, so we can just write c. Well then, that's our final answer and thank you for watching.

Evaluate the integrals in Exercises 33–52.∫ 8 cot⁴(t) dt

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Evaluate the integrals in Exercises 33–52.
∫ 8 cot⁴(t) dt