Evaluating definite integrals w/ trig

[instawin]

Hey everyone. I'm having trouble with the 2 attached definite integrals shown as a picture. I've also attached my attempt. My plan was to solve the indefinite integrals and then use fundamental theorem of calculus to evaluate them. I tried u-sub before going to integration by parts, but I couldn't get either to work. The picture of my work shows my attempt at IBP.

Teacher and wolfframalpha says it should evaluate to 0. I'm pretty stuck. Any help is greatly appreciated. Hoping that I'm overthinking this.

 

[Dr.Peterson]

Hey everyone. I'm having trouble with the 2 attached definite integrals shown as a picture. I've also attached my attempt. My plan was to solve the indefinite integrals and then use fundamental theorem of calculus to evaluate them. I tried u-sub before going to integration by parts, but I couldn't get either to work. The picture of my work shows my attempt at IBP.

Teacher and wolfframalpha says it should evaluate to 0. I'm pretty stuck. Any help is greatly appreciated. Hoping that I'm overthinking this.

Did you try a u-substitution, u = 1 - t? Do that in one of the definite integrals, and see what happens. And show us your work.

You are definitely overthinking (which really means underthinking and overdoing!)

Also, it may be a good idea (well, it always is) to show us the entire problem as given to you, so we can be sure what you are doing. I don't like problems that start with "...="!

 

[instawin]

Did you try a u-substitution, u = 1 - t? Do that in one of the definite integrals, and see what happens. And show us your work.

You are definitely overthinking (which really means underthinking and overdoing!)

Also, it may be a good idea (well, it always is) to show us the entire problem as given to you, so we can be sure what you are doing. I don't like problems that start with "...="!

I did try, but only with the indefinite versions. I was able to get them to cancel out when I did that substitution you suggested in one of the *definite* integrals; thank you for your suggestion/help. If people have a similar question I've attached Dr. Peterson's suggestion worked out.

edit: wasn't sure if pdfs were okay, so I remade it into a jpg

 

[Dr.Peterson]

Good work. I just renamed u back to t after the substitution (which is valid since the name of a variable is irrelevant), so that they became identical. That's equivalent to what you did, which amounts to making the "substitution" u = t in the second integral!

In general, if a problem is about definite integrals and I can't make progress working out an antiderivative, I go back to the definite integrals to see if there is anything special about them. Actually, in this case I guessed that would be the case without trying anything else, because I saw the symmetry in the integrands.