What is the integral from 0 to pi/2 of 8cos^2(θ) ?

The integral from 0 to pi/2 of 8cos^2(θ) is 2pi

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integral from 0 to pi/2 of 8cos^2(θ)

\int_{0}^{\frac{π}{2}} 8 cos^{2} (θ) d θ

2 π

Decimal Notation

6.28318 \dots

Solution steps

\int_{0}^{\frac{π}{2}} 8 cos^{2} (θ) d θ

Take the constant out:

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

= 8 \cdot \int_{0}^{\frac{π}{2}} cos^{2} (θ) d θ

Rewrite using trig identities

= 8 \cdot \int_{0}^{\frac{π}{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

= 8 \cdot \frac{1}{2} \cdot \int_{0}^{\frac{π}{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

= 8 \cdot \frac{1}{2} (\int_{0}^{\frac{π}{2}} 1 d θ + \int_{0}^{\frac{π}{2}} cos (2 θ) d θ)

\int_{0}^{\frac{π}{2}} 1 d θ = \frac{π}{2}

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

= 8 \cdot \frac{1}{2} (\frac{π}{2} + 0)

Simplify

8 \cdot \frac{1}{2} (\frac{π}{2} + 0) : 2 π

= 2 π

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What is the integral from 0 to pi/2 of 8cos^2(θ) ?
The integral from 0 to pi/2 of 8cos^2(θ) is 2pi