135. Evaluate ∫₀^(π/2) (sin x) / (sin x + cos x) dx in two way...

Question

135. Evaluate ∫₀^(π/2) (sin x) / (sin x + cos x) dx in two ways:

(a) By evaluating ∫ (sin x) / (sin x + cos x) dx, then using the Evaluation Theorem.

Accepted Answer

Hello there. Today we are 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. The integral evaluated from 0 to pi divided by 2 of cosine of x divided by sin of x plus cosine of x dx. OK. So it appears for this particular prompt, we're asked to ultimately evaluate the interval that is provided to us. So in order to evaluate this specific integral, we will need to recall and use the identity decomposition which states that cosine of x is equal to 1/2 multiplied by parentheses of. Sin x plus cosine of x plus parentheses of cosine of x minus sin of x. And we now need to substitute back into our integrand, so we need to take cosine of x divided by sin of x plus cosine of x, which will be equal to 1/2 multiplied by parentheses of. Sine of x plus cosine of x. Plus parentheses of cosine of x minus sine of x. All divided by. Sign of X. Plus Cosine of X. Fantastic. We are on a roll here. Which will be equal to when we write our final answer in its most simplified form. I'm going to skip a step. We will find that it's simply going to be equal to when we manipulate the expression to write in its most simple form, meaning we can't simplify it any further. We will find that it's equal to 1/2 multiplied by parentheses of 1 plus cosine of x minus sin of x. Divided by sin of x plus cosine of X. Fantastic. Thus, we can go ahead and write that the integral evaluated from 0 to pi divided by 2 of cosine of x divided by sin of x plus cosine of x. The x will be equal to 1/2 multiplied by the interval evaluated from 0 to pi divided by 2 of 1 plus cosine of x minus sin of x divided by sin of x. Plus cosine of x, DX will be equal to drum roll, 1/2 multiplied by the interval evaluated from 0 to pi divided by 2. Of dx plus 1/2 multiplied by the interval evaluated from 0 toi divided by 2 of cosine of x minus sine of x divided by sin of x plus cosine of x dx. And now, at this point, we need to integrate each term, starting with our first term, which we will be calling it, or we'll say that our first term is 1/2 multiplied by the interval evaluated from 0 to pi divided by 2 of DX. And this will be equal to 1/2 x evaluated from 0 to pi divided by 2, which will be equal to 1/2 multiplied by pi divided by 2 minus 0, which will be simply equal to pi divided by 4. And to integrate our second term, we need to use a substitution method. So let us state that you will be equal to sin of x plus cosine of x. Then that will mean that as a result, we could write that DU by DX will be equal to cosine of x minus sin of x, which will mean that DU will be equal to parentheses of. Cosine of x minus sin of x. In parentheses DX. So now we need to rewrite our limits. So X is equal to 0. So that will mean that you will be equal to sin of 0 plus cosine of 0, which will be equal to 0 + 1, which will be equal to 1. And when X is equal to pi divided by 2, that will mean that U is equal to sin of, so it's gonna be sin of pi divided by 2 plus cosine of pi divided by 2, which will be equal to 1 + 0, which will also be equal to simply 1. Fantastic. So now at this point, we need to recall that for any integral function, any function we can integrate for function F and any real number, A, we could write that the integral evaluated from A to B. Uh, so the integral evaluated from A to B of F of X, DX will be equal to 0. Thus, we can go ahead and write that 1/2 multiplied by the integral evaluated from 0 to pi divided by 2 of. Cosine of x minus sin of x divided by sin of x plus cosine of x dx. Will be equal to drum roll. 1/2 multiplied by the integral evaluated from 1 to 1 of duu divided by u, which is equal to 1/2 multiplied by 0, which will be equal to 0. Therefore, 1/2 multiplied by the integral evaluated from 0 to pi divided by 2 of dx plus 1/2 multiplied by the integral evaluated from 0 to pi divided by 2. Of cosine of x minus sine. Of X divided by, so this will be all divided by. Sign Of x plus cosine of x dx, which will be equal to pi divided by 4 plus 0, which will be equal to pi divided by 4, and that is our final answer. Hooray, we did it. So going back to look at the prom itself. We can safely say that our final answer for this evaluated integral is simply pi divided by 4, and that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

135. Evaluate ∫₀^(π/2) (sin x) / (sin x + cos x) dx in two ways:(a) By evaluating ∫ (sin x) / (sin x + cos x) dx, then using the Evaluation Theorem.

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135. Evaluate ∫₀^(π/2) (sin x) / (sin x + cos x) dx in two ways:
(a) By evaluating ∫ (sin x) / (sin x + cos x) dx, then using the Evaluation Theorem.