Evaluate the integrals in Exercises 33–52.
∫ sec³(x) tan...

Question

Evaluate the integrals in Exercises 33–52.

∫ sec³(x) tan³(x) dx

Accepted Answer

Welcome back everyone. Evaluate the integral of 2 cubed x tangent to the power of 5 x d x. Let's begin breaking down this function and specifically let's focus on tangents to the power of 5 of x. Basically what we can do is simply rewrite it as tangent to the power of 4 x multiplied by tangent x and using the properties of exponents that's tangent squared of x. Squared multiplied by tangent x. From here we're going to use trigonometric identities, right? Let's recall that tangent 2 is sequin 2 minus 1, so we can just write sequin 2 x minus 12 multiplied by Tangent acts. And now let's go ahead and. Rewrite our integral specifically that would be integral of sequent cubed x multiplied by tangent to the power of 5 x which is now. 22 x minus 12 multiplied by tangent x followed by dx. And now what we are going to do is simply use substitution. So for our substitution, let's set u equals sequent x. Let's remember that the next step is to find the derivative of u with respect to x, which is the derivative of sequent, and, and that's sequent x multiplied by tangent x. What we can do is simply rewrite sequent cubed as 2 square x multiplied by sequent x, right, which essentially will give us that term tangent x sequent x d x. Which is basically duu. We can multiply both sides by dx to show that the duu is equal to sequent x tangent x d x. And this allows us to rewrite the integrals integral off. Now, c 2 is U2, we're multiplying by U2 minus 12 and multiplying by DU. Now let's go ahead and square, so that be the integral of u 2 in parent u to the power of 4 minus 2 u2 plus 1 d u. And we can now distribute. So that'd be integral of u to the power of 6 minus 2 u. To the power of 4 plus e u2 d u and we can integrate using the power rule to get eu to the power of 7 divided by 7. -2 eu to the power of 5 divided by 5. Plus e u cubed divided by 3 plus e constant of integration c. And now since U is 2nd, we basically get 1/7. Second to the power of 7. Of x minus 2/5 Second to the power of 5 of x plus 1/3. 2 cubed of x. Plus a constant of integration c. And this is our final answer for this problem. Thank you for watching.

Evaluate the integrals in Exercises 33–52.∫ sec³(x) tan³(x) dx

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Evaluate the integrals in Exercises 33–52.
∫ sec³(x) tan³(x) dx