Evaluate the integrals in Exercises 33–52.
∫ sec⁴(x) tan...

Question

Evaluate the integrals in Exercises 33–52.

∫ sec⁴(x) tan²(x) dx

Accepted Answer

Welcome back everyone. Evaluate the integral sequence to the power of 6 of x tangent squared of xDx. For this problem before we begin, let's simply split sequent using the properties of exponents. We can write it as sequence to the power of 4 of x multiplied by sequin square x, and we're multiplying by tangent squared x. So now let's leave sequin 2 x for now and let's focus on sequence to the power of 4 of x. This can also be written as sequin 2 x 2. Followed by 2 squared x followed by tangent squared x. And now let's use trigonometric identities. 2 x can be written as 1 plus tangent 2 x. We're going to square it. And this is followed by. Tangent squared x 2 square x. Well then, and now let's rewrite the integral in terms of this expression. We're going to have one plus. Tangent squared x 2. Tangent squared x sequin square x d x. Now let's understand why this is useful. We're going to introduce U substitution. Let's suppose that u is equal to tangent x. And let's remember that the next step is to find the derivative of U with respect to x. There's a derivative of tangent x which is sequin squared x. And now, if we multiply both sides by dx, we can show that DU is equal to sequin squared XDX, right? This is exactly what we have within the integrant. And then we can rewrite the integral as 1 + EU 22, multiplied by U2, multiplied by the. We're going to square and distribute. Let's go ahead and do that. So that'd be 1 + 2 U 2 plus u to the power of 4 multiplied by U2DU. Now let's this tribute. We get integral. Of u2 + 2 u to the power of 4. Plus eu to the power of 6 d u. Well done. And now we can integrate using the power rules. So we get eu cubed divided by 3 for the first term. For the second term, we're going to factor out the constant and integrate u to the power of 4 using the power rule, so that'd be u to the power of 5 divided by 5. And for the final term again we're applying the power rules so that we use the power of 7 divided by 7. And now let's add the constant of integration c and finally replace you with tangent, so we get tangent. Cubed of x divided by 3. Plus 2 tangent to the power of 5 of x divided by 5. Plus Tangent to the power of 7 of x divided by 7. Plus c and that's our final answer for this problem. Thank you for watching.

Evaluate the integrals in Exercises 33–52.∫ sec⁴(x) tan²(x) dx

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Evaluate the integrals in Exercises 33–52.
∫ sec⁴(x) tan²(x) dx