7–84. Evaluate the following integrals.
49. ∫ tan³x · se...

Question

7–84. Evaluate the following integrals.

49. ∫ tan³x · sec⁹x dx

Accepted Answer

Welcome back, everyone. Evaluate the integral, integral of tangent cubed of X sequence to the power of 7 of XD X. For this problem we're going to use U substitution. Let's suppose that U is equal to C X. Now, if we identify the derivative DU divided by DX, this is going to be the derivative of sand, right, which is s and X tangent X. Now, specifically, if we focus on our integral, which is integral of tangent cubed of X sequence to the power of 7 of XD X. We noticed that we have tangent cubed, and in the expression of DU divided by DX we have tangent to the power of 1. So what we're going to do is simply express our integral as the product of tangent squared. We're going to split off one tangent, so our cube becomes a square, so we get tangent squared of X. We're going to split off 1 sequence, so sequence to the power of 7 becomes sequence to the power of 6. And then at the end we can just writes X, tangent X D X. Now notice that 2 X tangent XDX is simply DU, so we can labels extension XDX at the end as DU. Seeking to the power of 6 of X is you to the power of 6. Meaning our goal is to express tangent squared of x in terms of u, and we're going to use trigonometry. Let's recall that 2 squared of X is equal to 1 plus tangent squad of X, meaning tangent squared. Of X is equal to c 2x minus 1 or simply u2 minus 1. And that we can express tensions squared in terms of you as well, meaning our integral becomes. Integral of eu quad minus 1 multiplied by U to the power of 6 d u. And then we can distribute and evaluate two integrals. The first one will be U to the power of 8 because U2 multiplied by use the power of sexes. Us the power of 8. So we're integrating use the power of 8DDU. And subtracting the integral of you to the power of sex do you. We can apply the power rule that would be used to the power of 9 divided by 9. Minus u to the power of 7 divided by 7 plus A constant of integration C. And now we can substitutes and back, so we get 2 to the power of 9 of X. Divided by 9 minus 2 to the power of 7 of x. Divided by 7 plus a constant of integration C for an indefinite integral and that be our final answer. Thank you for watching.

7–84. Evaluate the following integrals.
49. ∫ tan³x · sec⁹x dx

Key Concepts

Integration Techniques

Trigonometric Identities

Definite vs. Indefinite Integrals

Watch next