• Support PF! Buy your school textbooks, materials and every day products Here!

Help with Trig Integral

  • Thread starter opus
  • Start date
  • #1
opus
Gold Member
704
130

Homework Statement


##\int_0^{π/8}sin^2(x)cos^2(x)##

Homework Equations




The Attempt at a Solution


Please see my attached work to see the train of thought. I've tried this thing about 100 times and still can't get the correct solution. I don't know if it's in the anti derivative evaluations of step (i) or the computation in step (ii)
 

Attachments

Answers and Replies

  • #2
Math_QED
Science Advisor
Homework Helper
2019 Award
1,354
487
I'm sorry to inform you that most members won't bother reading attached images. Since the image is hard to read (at least for me it is), all I can do is suggest you an easier appproach:

Observe that ##\sin(2x) = 2 \sin(x) \cos(x)##

Use this in your first step and your integral will boil down to (after a substitution) something like ##\int \sin^2(x) dx## which is a standard integral to solve.
 
  • #3
opus
Gold Member
704
130
Ok thanks!
 
  • #4
opus
Gold Member
704
130
Looks like it was computational. Marking as solved.
 
  • #5
Math_QED
Science Advisor
Homework Helper
2019 Award
1,354
487
Looks like it was computational. Marking as solved.
Did you find the correct answer using your own method/my method?
 
  • #6
opus
Gold Member
704
130
What I did was just redo the computations in the evaluation theorem. Still trying to see how to use your suggestion in the first step.
 
  • #7
Math_QED
Science Advisor
Homework Helper
2019 Award
1,354
487
What I did was just redo the computations in the evaluation theorem. Still trying to see how to use your suggestion in the first step.
##\sin^2 (x) \cos^2(x) = 1/4 \sin^2(2x)##
 
  • #8
opus
Gold Member
704
130
Are you squaring both sides?
 
  • #9
Math_QED
Science Advisor
Homework Helper
2019 Award
1,354
487
Are you squaring both sides?
##2\sin(x) \cos(x) = \sin(2x) \implies \sin(x) \cos(x) = 1/2 \sin(2x)##

and then indeed I square both sides.
 
  • #10
opus
Gold Member
704
130
Are you working to use the power reduction on the sin then?
 
  • #11
Math_QED
Science Advisor
Homework Helper
2019 Award
1,354
487
Are you working to use the power reduction on the sin then?
I use the standard identity ##\sin(2x) = 2\sin(x) \sin(x)##

This easily follows from ##\sin(a+b) = \sin(a) \cos(b) + \sin(b) \cos(a)## which is also a standard trig identity.

An easy proof can be given by writing down both sides of

$$e^{i(a+b)} = e^{ia}e^{ib}$$ using ##e^{ix} = \cos x + i \sin x## and comparing imaginary parts.

Knowing your trig identities can save a lot of time in such problems. Worth memorising imo.
 
  • #12
opus
Gold Member
704
130
Thats a good idea. Ive been mainly just doing what I can to drop the powers and get them into sums or differences.
 

Related Threads for: Help with Trig Integral

  • Last Post
Replies
6
Views
1K
  • Last Post
Replies
2
Views
795
  • Last Post
Replies
1
Views
995
Replies
2
Views
801
Replies
6
Views
685
Replies
3
Views
795
Replies
8
Views
2K
  • Last Post
Replies
2
Views
776
Top