What is the integral from 0 to 2 of cos^2(θ) ?

The integral from 0 to 2 of cos^2(θ) is 1+1/4 sin(4)

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integral from 0 to 2 of cos^2(θ)

\int_{0}^{2} cos^{2} (θ) d θ

1 + \frac{1}{4} sin (4)

Decimal

0.81079 \dots

Solution steps

\int_{0}^{2} cos^{2} (θ) d θ

Rewrite using trig identities

= \int_{0}^{2} \frac{1 + cos ( 2 θ )}{2} d θ

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int_{0}^{2} 1 + cos (2 θ) d θ

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int_{0}^{2} 1 d θ + \int_{0}^{2} cos (2 θ) d θ)

\int_{0}^{2} 1 d θ = 2

\int_{0}^{2} cos (2 θ) d θ = \frac{1}{2} sin (4)

= \frac{1}{2} (2 + \frac{1}{2} sin (4))

Simplify

\frac{1}{2} (2 + \frac{1}{2} sin (4)) : 1 + \frac{1}{4} sin (4)

= 1 + \frac{1}{4} sin (4)

What is the integral from 0 to 2 of cos^2(θ) ?
The integral from 0 to 2 of cos^2(θ) is 1+1/4 sin(4)