58. Two Methods Evaluate ∫(from 0 to π/3) sin(x) · ln(cos(x)) ...

Question

58. Two Methods Evaluate ∫(from 0 to π/3) sin(x) · ln(cos(x)) dx in the following two ways:

b. Use substitution.

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Evaluate the integral from 0 to pi divided by 6 of cosine of X multiplied by the natural log of sin of XDX using substitution. Awesome. So it appears for this particular problem we're asked to take the integral that's provided to us and we're asked to evaluate it using substitution. So now that we know what we're ultimately trying to solve for and noting once again that we are asked to use substitution, our first step that we need to take is we need to recall and use the substitution method. So using the substitution method, let us state that W. is equal to sin of x, which will mean that DW is equal to cosine of xDX. Awesome. Now we need to change the limits of integration. So when X is equal to 0, that will mean that W is equal to sin of 0, which is equal to 0. And when X is equal to pi divided by 6, that will mean that W is equal to sign of pi divided by 6, which will be equal to 1/2. So therefore, as we should recall, the integral will therefore become the integral from 0 to pi divided by 6 of Cosine of x multiplied by the natural log of the sine of X DX will be equal to the integral from 0 to 1/2 of the natural log of WDW. Awesome. So, With that in mind, our next step now is we need to, this is this once again will be our second step. We now need to recall and use the acronym Lipi to help us choose the values of you and DV. So let us note that Lipi stands for L I P E T, where L stands for logamithic functions such as the natural log of X. I stands for inverse trigonometric, such as inverse tangent of X. And P stands for polynomial, and P for polynomial could be a value like x2 or Y, etc. E stands for exponential such as E to the power of X, and finally T stands for trigometrics such as sin. X or cosine of X, etc. So once again we're using this acronym Lipi to help us determine our values of. So once again we're using it to choose values of you and DV. So with that in mind, using lipit, we will note that you will be equal to the natural log of W. And the natural log of W is going to be equal to you, and DV will be equal to 1 multiplied by DW. So now we need to compute, so we need to note that U is equal to 1 divided by WDW and that V is equal to W. So now with that in mind, our third step is we need to recall and apply integration by parts. So recalling and using integration by parts, we could say that the integral of the natural log of WDW is equal to W multiplied by the natural log of W minus the integral of W multiplied by 1 divided by WDW. So now we need to we need to simplify the second integral, and when we simplify the second integral, we will find that it's equal to W multiplied by the natural log of w minus the integral of 1DW, which is equal to W multiplied by the natural log of W minus W plus C, where C denotes some constant value. Awesome. Our fourth step now is we need to evaluate the definite integral. So with that in mind, we can go ahead and write the integral from 0 to 12 of the natural log of W. DW is equal to square brackets of W multiplied by the natural log of W minus W evaluated from once again noting it's in square brackets, evaluated from 0 to 12. Fantastic. And we need to note that at W equals 12, we can go ahead. And write that 1/2 multiplied by the natural log of 1/2 minus 1/2 is equal to 1/2 multiplied by the parenthesis of negative natural log of 2. Minus 12, which will be ultimately all equal to -12 multiplied by the natural log of 2 minus 1/2. Fantastic. So, moving right along here. Our next step is we need to consider, or part of step 4, our next thing we need to solve for is that at w equals 0. So at W equals 0 we can state once again focusing at W equals 0, we could state that the limit. So we can say that the limit. As W approaches 0 from the right of W multiplied by the natural log of W will be equal to 0, and we can state the limit as W approaches 0 from the right of W will be equal to 0. So that will mean that we can therefore write that the limit as W approaches 0 from the right. And just in case you're wondering, this positive symbol and the power denotes that it's approaching from the right, so that's where we get as W approaches 0 from the right. So the limit as W approaches 0 from the right of parentheses W multiplied by the natural log of W minus W will be equal to 0. So that means we could go ahead and write that negative 1/2 multiplied by the natural log 2, minus 12 will be equal to -12 multiplied by parentheses 1 plus the natural log of 2. Fantastic. Don't forget that it's written inside a parentheses, and that will be our final answer. Hooray, we did it. So going back to the prom itself to recap what we've learned, that will mean that our final answer, without a doubt once again will have to be. Negative, so -12. Multiplied by parentheses 1 plus the natural log of 2, and that's it. That is our final answer. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

58. Two Methods Evaluate ∫(from 0 to π/3) sin(x) · ln(cos(x)) dx in the following two ways:
b. Use substitution.

Key Concepts

Integration by Substitution

Properties of Logarithmic and Trigonometric Functions

Definite Integrals and Limits of Integration

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