Evaluating integrals Evaluate the following integrals. &n...

Question

Evaluating integrals Evaluate the following integrals.

∫ d𝓍/[(tan⁻¹ 𝓍) (1 + 𝓍²)]

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 DX divided by the quantity of 1 + 4 X squared in quantity multiplied by the inverse tangent of 2 X in quantity. So for this problem, we just need to evaluate our integral, and we're given the integral of DX divided by the quantity of 1 + 4 X squared. In quantity multiplied by the inverse tangent of 2 X in quantity. So, in order to evaluate this integral, we're actually going to use substitution. So, the first thing we'll do is set U equal to the inverse tangent. Of 2 X. And so now we need to take the derivative find DU. So DU is going to be equal to, and when I take the derivative with respect to X of the inverse tangent of 2 X, I get 1 divided by the quantity of 1 plus the quantity of 2 X in quantity squared in quantity. Then I'll need to multiply this by the derivative of 2 X with respect to X by applying our chain rule, and that's just gonna be 2. And then this gets multiplied by DX. So we can simplify this expression, and we'll get that DU is equal to 2. Divided by the quantity of 1 +4 X squared in quantity multiplied by DX. So now if we look at our interval, we see that we have the quantity of DX divided by the quantity of 1 + 4 X squared in quantity. So to match that up with our expression for DU, we'll just need to divide both sides of that expression by 2. In doing so, we will have 1/2 multiplied by DU is equal to DX divided by the quantity of 1 + 4 X squared in quantity. So now we can do our substitution into our integral. In doing so, we get that this is equal to the integral of 1 divided by U, multiplied by 1/2 multiplied by D U. And we can actually evaluate this integral in terms of you. So this is going to be equal to. The one half is a constant, so I can pull it outside the integral. So I'll have 1/2, and what I'm left with is the integral of DU divided by U. And recall that that is going to be equal to the natural log of the absolute value of U. Then we'll also have to have our integration constant, so we'll have a plus C as well. Now all we need to do is substitute in for you, the inverse tangent of 2 X. In doing so, we see that this is going to be equal to 1/2 multiplied by the natural log of the absolute value of the quantity of the inverse tangent. Of 2 X. In quantity Plus D. And this here is the answer that we were looking for for this problem. So I hope this explanation has been useful, and I'll see you in the next video.

Evaluating integrals Evaluate the following integrals.

∫ d𝓍/[(tan⁻¹ 𝓍) (1 + 𝓍²)]

Key Concepts

Integration

Inverse Trigonometric Functions

Differentiation of Composite Functions

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