Evaluate the integrals in Exercises 37–44.
∫ tan³(x) sec...

Question

Evaluate the integrals in Exercises 37–44.

∫ tan³(x) sec³(x) dx

Accepted Answer

Welcome back everyone. Evaluated the following integral. Integral of tension squared x sequence to the power of 4 x dx. Before we begin integrating, let's focus on the function that we're integrating. That's tangent squared x sequent to the power of 4 x, right? And this function is not elementary. So what we want to do is simply use trigonometric identities to simplify and then possibly use use substitution. What we can do is simply split sequin into sequin 2 multiplied by sequin 2, so that'll be tangent 2 x multiplied by sequin 2 x multiplied by another sequin 2 x. And now let's remember trigonometric identities. 22 x can be written as 1 plus tangent 2 x. So now we can rewrite the integrals. Integral of tangent squared x multiplied by sequin 2 x, which is 1 + tangent 2 x, followed by sequent square x. Notice how we are using the identity on one sequin squared, but we're leaving the other sequin squared x as it is, and soon we will understand why this is useful. Well, if we apply you substitution and we let u be equal to tangent x. We will notice that the derivative of you with respect to x is the derivative of tangent x, right? Which is c square x, and if we multiply both sides by dx, we will show that d u is square x d x. This is exactly what we have within the integral, right? We have sequin square x dx, which now becomes DU. And tension square x becomes U2. So now we can rewrite the integral in terms of UN DU. That would be the integral of U2 multiplied by 1 + u2 DU. Distributing We are going to integrate u2 plus u to the power of 4DU. And now we can apply the power rule that will be u cubed divided by 3. Plus eu to the power of 5 divided by 5. Plus a constant of integration c. And now our final step is to replace you with tangent. So we get tangent, cubed. Of x divided by 3. Plus tangent raise to the power of 5 of x. Divided by 5 plus c, and that's our final answer for this problem. Thank you for watching.

Evaluate the integrals in Exercises 37–44.∫ tan³(x) sec³(x) dx

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Evaluate the integrals in Exercises 37–44.
∫ tan³(x) sec³(x) dx