Calculate int_0^(2pi) cos^2x .dx∫2π0cos2x.dx?
A) Zero
B) 2
C) 1/2
D) piπ
The answer is D for your reference
A) Zero
B) 2
C) 1/2
D)
The answer is D for your reference
2 Answers
Yes, the correct answer is
Explanation:
Use the power reduction formula
cos^2x= (1 + cos2x)/2cos2x=1+cos2x2 .
So we have:
I = int_0^(2pi) (1 + cos2x)/2I=∫2π01+cos2x2
I = 1/2int_0^(2pi) 1 + cos(2x)dxI=12∫2π01+cos(2x)dx
I = 1/2int_0^(2pi) 1 dx + 1/2int_0^(2pi) cos(2x)dxI=12∫2π01dx+12∫2π0cos(2x)dx
If we let
I = 1/2[x]_0^(2pi) + 1/2(1/2)[sin(2x)]_0^(2pi)I=12[x]2π0+12(12)[sin(2x)]2π0
The integral equals
I = 1/2(2pi) + 1/4(sin(4pi) - sin(0))I=12(2π)+14(sin(4π)−sin(0))
I = pi - 1/4(0 - 0)I=π−14(0−0)
I = piI=π
Answer
Hopefully this helps!
We can rewrite this:
Note that
Rewrite
Add
Evaluate these integrals:
Dividing by