7–84. Evaluate the following integrals.
38. ∫ from π/6 t...

Question

7–84. Evaluate the following integrals.

38. ∫ from π/6 to π/2 [cos x · ln(sin x)] dx

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to evaluate the interval from 0 to pi divided by 2 of sin of X, multiplied by the natural log of cosine of XDX. We're given 4 possible choices as our answers. For choice A, we have pi divided by 2. Choice B, we have 1, for choice C, we have 0. for choice D, we have minus 1. So in order to evaluate this integral, we'll actually have to use integration by parts. The first thing we need to do is identify our U and DV. So, for you, we're gonna set that equal to the natural log of cosine of X. And so that means that DU is going to be equal to the derivative with respect to X of U, which is going to be the derivative of our natural log of cosine of X with respect to X. And recall that when you take the derivative of the natural log function, you get 1 divided by its argument. So we'll get 1 divided by cosine of X, we'll need to apply our chain rule and multiply this by the derivative of our argument with respect to X. So when we take the derivative with respect to X of cosine of X, we get minus sine of X. And then that gets multiplied by DX. But we also need to identify DV, which is going to be what's left in our integral, which is going to be sin of X D X, and we need to integrate this in order to get V. And so when you integrate sin of X with respect to X, we see that V is going to be equal to minus cosine of X. So, we can now evaluate our integrals, and so we're given the integral from 0 to pi divided by 2 of sin of X. Multiplied by the natural log of cosine of X. DX And this is going to be equal to we here recall how you integrate by parts. So the first term it's gonna be a boundary term, which is going to be you multiplied by V, which is going to be the natural log of cosine of X, multiplied by minus cosine of X. And that is evaluated at our lower limit of 0 and upper limit of high divided by 2. And then we'll have minus the integral from 0 to pi divided by 2. Of VDU. So that's going to be minus cosine of X. Multiplied by the quantity of minus sine of X. Divided by cosine of X in quantity, DX. And so, evaluating our boundary term here, we see that it's going to be equal to the natural log of cosine of pi divided by 2. Multiplied by minus cosine of pi divided by 2. Then we'll have minus the natural log of cosine of zero, multiplied by minus cosine of 0. Then we'll have minus the integral from 0 to pi divided by 2, and here we can simplify the second integral. So the cosine terms will cancel out, and the two negative signs will multiply together to give us a positive sign. So inside the interval, we just have sin of Xs. DX Now, if we look at the first term in our result here, we see that we run into a little bit of an issue. We're trying to get a single number. Notice that the cosine of pi divided by 2 is equal to 0. So we have the natural log of 0 multiplied by 0. However, the natural log of zero approaches minus infinity. So we really have minus infinity multiplied by 0, which is an indeterminate form. So we'll actually have to use limits to see what value that this expression actually approaches it approaches as X approaches pi divided by 2. So we're gonna be looking at the limit. As X approaches pi divided by 2, and we actually want to rewrite our boundary term in a manner that will allow us to use Lobital's rule. So we're going to rewrite it as the natural log of cosine of X. Divided by The quantity of 1 divided by minus cosine of X in quantity. And so, if we evaluate this limit by direct substitution, We see that in the numerator, we will get minus infinity, but in the denominator, we'll also have minus infinity. So we got minus infinity divided minus infinity, which is again an indeterminate form, but it is a form in which we can use Lopital's role. So, evaluating this limit, using Lopital's rule, this is going to be equal to the limit, as X approaches pi divided by 2, and here we call to apply Lopital's rule, we just need to take the derivative with respect to X of both our numerator and denominator of our expression. So, when we take the derivative of our numerator, we're taking the derivative with respect to X of the natural log of cosine of X, and we actually found this earlier when we were calculating DU, and this is actually minus sine of X divided by cosine of X. And then this quantity is going to be divided by the derivative with respect to X of 1 divided by minus cosine of X. So evaluating that derivative, we get one divided by. Cosine squared of X multiplied by minus sine of X. So simplifying this expression, this is going to be equal to the limit. As X purchase pi divided by 2. Of cosine of X. And here, when we evaluate this limit, by direct substitution, we just get the cosine of pi divided by 2, which is equal to 0. So that means that the first term in our previous expression is just equal to 0. That means that are integral. From 0 to pi divided by 2 of sin of X, multiplied by the natural log of cosine of X. DX It's equal to 0 minus, and here for the next term, we have the natural log of cosine of 0. Minus, or sorry, multiplied by minus cosine of zero. So recall that cosine of 0 is equal to 1, and so we'll have the natural log of 1 multiplied by -1, but the natural log of 1 is 0, so this term is actually 0 as well. Then we'll have minus the integral from 0 to pi divided by 2 of sin of XDX. And so we just need to integrate sin of XDX, which is equal to minus cosine of X, and this needs to be evaluated at the end points of 0 and pi divided by 2. So this is going to be equal to. Cosine of pi divided by 2 minus cosine of 0. And again, cosine of pi divided by 2 is equal to 0, and cosine of 0 is equal to 1. That means that this expression is going to be equal to -1. So this here is the answer that we were looking for for this problem. And if we compare that to our answer choices, we see that the correct answer choice corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

7–84. Evaluate the following integrals.
38. ∫ from π/6 to π/2 [cos x · ln(sin x)] dx

Key Concepts

Definite Integrals

Integration by Parts

Logarithmic and Trigonometric Functions

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