please help! Im trying to integrate (Sin^2) (x)*(Cos^2)(x) from 0 to pi/2 the answer says pi/16 but im not getting that answer... please show your work and explain what identities youve used im trying to use sinxcosx = 1/2 (sin 2x) but im not sure if this is correct.
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integration of trig functions
- Thread starter Caitlyn
- Start date
You can go that way but will need to continueplease help! Im trying to integrate (Sin^2) (x)*(Cos^2)(x) from 0 to pi/2 the answer says pi/16 but im not getting that answer... please show your work and explain what identities youve used im trying to use sinxcosx = 1/2 (sin 2x) but im not sure if this is correct.
sin2(x) cos2(x)=\(\displaystyle \frac{1}{4}\) sin2(2x)
=\(\displaystyle \frac{1}{4}\) [1-cos2(2x)]
=\(\displaystyle \frac{1}{4}\) {1-\(\displaystyle \frac{1}{2}\)[cos(4x)+1]}
etc.
Thanks this is really helping, i have another question though. When I distribute the 1/4 i have the subtraction of a constant. not necessarily in this problem but in others im running into, i have to do a substitution where there is the addition or subtraction of a constant and im forgetting what to do with it. can i pull the constant out front and say 1/4 - 1/8 integral 1/2 [cos(4x)+1] ? im sorry if this question is confusing its hard to word exactly what im having trouble understanding. Basically when im doing a substitution i dont know how to get rid of constants when they are being added or subtracted not just as a multiplier.
[FONT=MathJax_Main]1/[/FONT][FONT=MathJax_Main]4[/FONT] {1-[FONT=MathJax_Main]1/[/FONT][FONT=MathJax_Main]2[/FONT][cos(4x)+1]}
[FONT=MathJax_Main]1/[/FONT][FONT=MathJax_Main]4[/FONT] {1-[FONT=MathJax_Main]1/[/FONT][FONT=MathJax_Main]2[/FONT][cos(4x)+1]}
You still have to integrate the constants, but they can be pulled out in front of the integral. So, to continue with where we wereThanks this is really helping, i have another question though. When I distribute the 1/4 i have the subtraction of a constant. not necessarily in this problem but in others im running into, i have to do a substitution where there is the addition or subtraction of a constant and im forgetting what to do with it. can i pull the constant out front and say 1/4 - 1/8 integral 1/2 [cos(4x)+1] ? im sorry if this question is confusing its hard to word exactly what im having trouble understanding. Basically when im doing a substitution i dont know how to get rid of constants when they are being added or subtracted not just as a multiplier.
[FONT=MathJax_Main]1/[/FONT][FONT=MathJax_Main]4[/FONT] {1-[FONT=MathJax_Main]1/[/FONT][FONT=MathJax_Main]2[/FONT][cos(4x)+1]}
sin2(x) cos2(x)= [FONT=MathJax_Main]1/[/FONT][FONT=MathJax_Main]4[/FONT] {1-[FONT=MathJax_Main]1/[/FONT][FONT=MathJax_Main]2[/FONT][cos(4x)+1]}
=[FONT=MathJax_Main]1/[/FONT][FONT=MathJax_Main]4[/FONT] [1-[FONT=MathJax_Main] (1/[/FONT][FONT=MathJax_Main]2[/FONT]) cos(4x) - (1/2)]
=[FONT=MathJax_Main]1/[/FONT][FONT=MathJax_Main]4[/FONT] [(1/2) -[FONT=MathJax_Main] (1/[/FONT][FONT=MathJax_Main]2[/FONT]) cos(4x)]
=1/8 [ 1 - cos(4x)]
and we have
\(\displaystyle \int_0^{\frac{\pi}{2}} sin^2(x) cos^2(x) dx\)
=\(\displaystyle \int_0^{\frac{\pi}{2}} \frac{1}{8} [1- cos(4x)] dx\)
=\(\displaystyle \frac{1}{8}\int_0^{\frac{\pi}{2}} [1- cos(4x)] dx\)
=\(\displaystyle \frac{1}{8}[\, \int_0^{\frac{\pi}{2}} dx - \int_0^{\frac{\pi}{2}} cos(4x)dx\, ]\)