Calculate int_0^(2pi) cos^2x .dx2π0cos2x.dx?

A) Zero
B) 2
C) 1/2
D) piπ
The answer is D for your reference

2 Answers
Sep 5, 2017

Yes, the correct answer is DD, or piπ.

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=122π01+cos(2x)dx

I = 1/2int_0^(2pi) 1 dx + 1/2int_0^(2pi) cos(2x)dxI=122π01dx+122π0cos(2x)dx

If we let u = 2xu=2x in the second integral, then 1/2du = dx12du=dx. The integral of cosucosu is sinusinu and reverse your substitution to get sin(2x)sin(2x).

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=π14(00)

I = piI=π

Answer DD, as required.

Hopefully this helps!

Sep 6, 2017

int_0^(2pi)cos^2(x)dx2π0cos2(x)dx

We can rewrite this:

int_0^(2pi)cos^2(x)dx=int_0^(2pi)(cos^2(x)-sin^2(x)+sin^2(x))dx2π0cos2(x)dx=2π0(cos2(x)sin2(x)+sin2(x))dx

Note that cos(2x)=cos^2(x)-sin^2(x)cos(2x)=cos2(x)sin2(x):

int_0^(2pi)cos^2(x)dx=int_0^(2pi)cos(2x)dx+int_0^(2pi)sin^2(x)dx2π0cos2(x)dx=2π0cos(2x)dx+2π0sin2(x)dx

Rewrite sin^2(x)sin2(x) using sin^2(x)+cos^2(x)=1sin2(x)+cos2(x)=1:

int_0^(2pi)cos^2(x)dx=int_0^(2pi)cos(2x)dx+int_0^(2pi)(1-cos^2(x))dx2π0cos2(x)dx=2π0cos(2x)dx+2π0(1cos2(x))dx

int_0^(2pi)cos^2(x)dx=int_0^(2pi)cos(2x)dx+int_0^(2pi)dx-int_0^(2pi)cos^2(x)dx2π0cos2(x)dx=2π0cos(2x)dx+2π0dx2π0cos2(x)dx

Add int_0^(2pi)cos^2(x)dx2π0cos2(x)dx to both sides of the equation:

2int_0^(2pi)cos^2(x)dx=int_0^(2pi)cos(2x)dx+int_0^(2pi)dx22π0cos2(x)dx=2π0cos(2x)dx+2π0dx

Evaluate these integrals:

2int_0^(2pi)cos^2(x)dx=|1/2sin(2x)|_0^(2pi) +x|_0^(2pi)22π0cos2(x)dx=12sin(2x)2π0+x2π0

2int_0^(2pi)cos^2(x)dx=1/2(sin(4pi)-sin(0))+(2pi-0)22π0cos2(x)dx=12(sin(4π)sin(0))+(2π0)

2int_0^(2pi)cos^2(x)dx=2pi22π0cos2(x)dx=2π

Dividing by 22:

int_0^(2pi)cos^2(x)dx=pi2π0cos2(x)dx=π