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

The integral from pi/2 to pi of cos^2(θ) is pi/4

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integral from pi/2 to pi of cos^2(θ)

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

\frac{π}{4}

Solution steps

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

Rewrite using trig identities

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

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

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

= \frac{1}{2} (π - \frac{π}{2} + 0)

Simplify

\frac{1}{2} (π - \frac{π}{2} + 0) : \frac{π}{4}

= \frac{π}{4}

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What is the integral from pi/2 to pi of cos^2(θ) ?
The integral from pi/2 to pi of cos^2(θ) is pi/4