65-68. Reduction formulas Use the reduction formulas in a tabl...

Question

65-68. Reduction formulas Use the reduction formulas in a table of integrals to evaluate the following integrals.

67. ∫tan⁴(3y) dy

Accepted Answer

Hello. In this video, we are going to be evaluating the following integral using the appropriate reduction formula. We are given the integral of tangent to the 6 of 4 ZD Z. Now, the first thing we can do in order to simplify this integral is to apply a U substitution. We will go ahead and allow you to equal to 4 Z, and that means D U will equal to 4DZ. Since we're looking for a substitution of DZ, we can further rewrite the statement to be 14th DU is equal to DZ, and by applying these substitutions, we can rewrite the given integral to be 14th multiplied by the integral of tangent to the 6th power of U D U. Now, in order to approach this type of problem, we can go ahead and simplify this problem by using the reduction formula for tangent to the nth power. That formula is defined as the following. If we have the integral of tangent to the nth power XDX, then the reduction formula will be 1 divided by N minus 1, multiplied by tangent of N minus 1 of X. Minus the integral of tangent of the power N minus 2 of XDX, and we are going to apply this formula where N is equal to 6. So by using this power reduction formula, we can rewrite the integral to be 1/4 multiplied by 1 divided. By 5, multiply by tangent to the 5th power of X minus the integral of tangent to the 4th power of XDX. Now, we can further simplify this integral by applying the reduction formula one more time to tangent to the 4th power of X. That is going to allow us to rewrite everything to be 1/4, multiplied by 1/2, tangent to the 5th power of X minus the reduction formula for tangent to the 4th power of X. That is going to be 1/3. Tangent cubed of X minus the integral of tangent squared. Of XDX. Now, let's go ahead and simplify this by evaluating tangent squared of X. Now, the integral of tangent squared of X can be rewritten as the integral of sequent squad of X minus 1. Now, this is just us rewriting or simplifying by using the Pythagorean identity for tangent squared, but that is going to evaluate to be tangent of X minus X. So, that is going to allow us to further simplify everything to be 1/4, multiplied by 1/2, tangent to the 5th power of X minus the quantity 1/3 tangent cubed of X. Minus the quantity tangent of X minus X. And because we were working with the indeterminate integral originally, we will add a generic constant C. So, let's go ahead and simplify the given quantity. We will start by distributing -1 into tangent of X minus X. This is going to allow us to simplify that quantity to be negative tangent of X plus X. Then what we are going to do is we are going to distribute -1 to each term within this quantity. That is going to allow us to rewrite this as 1/4, multiplied by 1/5 tangent to the 5th power of X. Minus 1/3. Tension cubed of X. Plus tangent of X minus X plus C. And finally we are going to distribute 14th to everything within the quantity. That is going to give us 1 divided by 20 tangent to the 5th power of X. Minus 1 divided by 12, tangent cubed X. Plus 14th tangent of X. Minus 1/4 X plus C. Now, recall earlier in the problem, we did perform a U substitution, and what that means is that everything here is currently in terms of you. And with that being said, what we need to do now in order to finish the problem is we just need to reapply our original U substitution. Recall that we allowed U to equal to 4 Z. And so if we resubstitute U in the terms of X, that will allow us to simplify the answer to be 1 divided by 20, multiplied by tangent to the 5th power of 4 Z minus 1/12 multiplied by tangent cubed. Of 4 Z Plus 14th multiplied by tangent of 4 Z. Minus 1/4 multiplied by 4 Z, which is just a Z plus C, and this is going to be the solution to the original integral that was given to us. And with that being said, the overall solution to this problem is going to be D. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

65-68. Reduction formulas Use the reduction formulas in a table of integrals to evaluate the following integrals.
67. ∫tan⁴(3y) dy

Key Concepts

Reduction Formulas

Integration of Trigonometric Functions

Substitution Method

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