9–61. Trigonometric integrals Evaluate the following integrals...

Question

9–61. Trigonometric integrals Evaluate the following integrals.

34. ∫ tan⁹x sec⁴x dx

Accepted Answer

Welcome back, everyone. Evaluate the integral integral of tangent to the power of 5 of X, sequence to the power of 6 of XD X. So, for this problem, first of all, we can express sequent to the power of 6 of X assent. To the power of 4 of X multiplied by 2 to the power of 2 of X, we want to isolates 2 X, right? And then we're going to use one of our identities in trigonometry. C2 X is equal to 1 + tangent squared X. And now let's introduce you substitution. Let's suppose that U is equal to tangent of X, right? We can recall that the derivative DU divided by DX is squad of X. So now we can show that the U is equal to c 2d XD X. And from here we are going to consider our original integral, right? So we had integral of tangent to the power of 5 of X multiplied by 2 to the power of 4 of X multiplied by sequent squared XD X. So now let's notice that we have squared XD X. This is going to be our DU. We have tensions to the power of 5 of X. This is going to be U to the power of 5 and what is 2 to the power of 4. Well, essentially, if c 2 x is 1 + tangent squared X, which is 1 + u squad, then 2. So the power of 4 of X is equal to 1 plus a U2 all of that squared. And now if we apply the formula and square it, we're going to get 1 + 2 U 2 + U to the power of 4. So we have also our expression forsakens the power of 4 of X in terms of u that be 1 plus. To you Squared plus u to the power of 4. Therefore, our integral becomes integral of U to the power of 5, multiplied by 1 + 2 U 2 + u to the power of 4. The EU And that we can distribute and we end up with 3 integrals use the power of 5DU. Plus 2 integral to the power of 7 DU plus integral of U to the power of 5 + 4, that's 9. The EU Let's apply the power rule. For the first integral, we get you the power of 6. Divided by 6. Plus 2 multiplied by the integral of uses the power of 7 is the power of 8 divided by 8. And finally, plus he used the power of 10 divided by 10. Plus a constant of integration C. And now since U is tangent, we can substitute U equals tangent X and begin with the term with the highest exponent, so that be tangent to the power of 10 of X divided by 10. Plus 2 divided by 8 is 4, so we get plus tangent to the power of 8 of X divided by 4 plus. Tangent to the power of 6 of X. Divided by 6. Plus a constant of integration C and that would be our final answer for this problem. Thank you for watching.

9–61. Trigonometric integrals Evaluate the following integrals.
34. ∫ tan⁹x sec⁴x dx

Key Concepts

Trigonometric Functions

Integration Techniques

Secant and Tangent Identities

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