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

Question

9–61. Trigonometric integrals Evaluate the following integrals.

31. ∫ 20 tan⁶x dx

Accepted Answer

Hello. In this video, we are going to be solving the following integral. We are giving the integral of 15 multiplied by tangent to the 4th power of XDX. Now, the first thing we can do is we can go ahead and move that 15 as a coefficient in front of the integral. That will leave us with 15 multiplied by tangent to the 4th power of XDX. Now, since we have tangent to the 4th power, what we can do is we can simplify this integral by applying the power reduction formula for tangent to the 4th power that is given as the following. If we have an integral of tangent to the nth power of X, then this will simplify to 1 divided by n minus 1, multiplied by tangent to the power of n minus 1 X minus the integral of tangent. To the power of N minus 2 of XDX. This can be applied when we have an original exponent that is greater than or equal to 2, and in this case here, we have a value of the exponent where N is equal to 4. So if we were to apply this reduction formula to the tangent to the 4th power, we can rewrite this integral to be 15, multiplied by 1 divided by 4 minus 1, which is 3, multiplied by tangent to the 4 minus 1 power, which is 3, X minus the integral of tangent to the power of N minus 2, which will leave us with tangent squared of XDX. Next, what we're going to do is we're going to distribute that 15 to each term within the group. So that will leave us with 15 divided by 3, which is 5 tangent cubed of X. -5 multiplied by the integral of tangent squared of XDX. We can solve this integral by using a Pythagorean identity. If we were to rewrite the integral of tangent squared of X by using the Pythagorean identity, we can represent this integral as the integral of squad of X minus 1d X, and if we were to take the antiderivative of both terms, we will be left with tangent of X minus X. So that is going to simplify the second half of this quantity. And we will be left with 5 multiplied by tangent cubed of X, minus 5 multiplied by the quantity, tangent of X minus X, and because we were working with the general integral, we will add a general constant C. Finally, we are going to distribute that -5 to tangent of X minus X, and we will be left with 5 multiplied by tangent cubed of X minus 5 multiplied by tangent of X plus 5 X plus C. and this is going to be the simplest form of the solution to the integral. And with that being said, the overall solution is going to bed. 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.

9–61. Trigonometric integrals Evaluate the following integrals.
31. ∫ 20 tan⁶x dx

Key Concepts

Trigonometric Functions

Integration Techniques

Power Reduction Formulas

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