2–74. Integration techniques Use the methods introduced in Sec...

Question

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

22. ∫ tan³ 5θ dθ

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to evaluate the integral, and we're given the integral of tangent cubed of 4 XDX. So we need to evaluate this integral, and the first step is to rewrite our interval. So we'll have the integral, and here we rewrite this as tangent of 4 X, multiplied by tangent square to 4 X, but we can rewrite tangent square of 4 x as the quantity of sequence squared. Of 4 X. 1 in quantity. So here we just use our trig identities to do that. And then that gets multiplied by DX. So, we now want to do is to distribute our tangent of 4X, and to rewrite this as two separate integrals. This is going to be equal to. The interval of tangent of 4 X multiplied by sequin squared of 4 X. DX? Minus the interval of tangent of 4 X. DX. So, we want to do now is to look at each one of these intervals individually. So starting with the first integral, we'll have the integral of tangent of 4 X. Multiplied by sequin squared. Of 4 X. DX And to do this integration, we're actually going to use U substitution. So we're gonna set U equal to tangent of 4 X. So that means that DU is going to be equal to 4 multiplied by sin. Squared of 4 X. DX. Now, if I look at my integral here, we see that we just have sequence squad of 4 XDX in our integral. So that means we'll need to divide both sides of this equation by 4, and we'll get 1 divided by 4 multiplied by DU, is equal to sequence squad of 4 X. DX. So making the substitutions, we see that our interval is going to be equal to. The integral of U multiplied by 1 divided by 4 multiplied by D U. And so, performing this integration, this is going to be equal to. We'll factor out the 1/4 in front of the integral, so we'll have 1 divided by 4, and then that gets multiplied by the integral of you with respect to you. And recall that when we integrate that, we get the quantity of U squared divided by 2, and then we'll have to add an integration constant to this. We'll add plus C. So now what we can do is simplify this expression, and go ahead and substitute back in tangent of 4 X for you. In doing so, we see that this is going to be equal to 1 divided by 8, multiplied by the tangent squared of 4 X plus C. So, now, if we look at the 2nd interval that we have, we have the interval of tangent of 4 XDX. And recall that this is going to be equal to 1 divided by 4, multiplied by the natural log of the absolute value of the quantity of sequent of 4 X. In quantity Plus C. So, now we can combine both of these results to get our final answer. So, we'll have the interval of tangent cubed of 4 X. DX Is going to be equal to. 1 divided by 8 multiplied by tangent squared of 4 X. -1 divided by 4, multiplied by the natural log of the absolute value of the quantity of sequent of 4 X. In quantity Plus C, where we have combined our two integration constants into just a single overall integration constant. So this here is the answer that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
22. ∫ tan³ 5θ dθ

Key Concepts

Integration Techniques

Trigonometric Identities

Substitution Method

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