What is the integral from pi/4 to pi/2 of sin^2(x) ?

The integral from pi/4 to pi/2 of sin^2(x) is (pi+2)/8

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integral from pi/4 to pi/2 of sin^2(x)

\int_{\frac{π}{4}}^{\frac{π}{2}} sin^{2} (x) d x

\frac{π + 2}{8}

Decimal

0.64269 \dots

Solution steps

\int_{\frac{π}{4}}^{\frac{π}{2}} sin^{2} (x) d x

Rewrite using trig identities

= \int_{\frac{π}{4}}^{\frac{π}{2}} \frac{1 - cos ( 2 x )}{2} d x

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}{4}}^{\frac{π}{2}} 1 - cos (2 x) d x

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}{4}}^{\frac{π}{2}} 1 d x - \int_{\frac{π}{4}}^{\frac{π}{2}} cos (2 x) d x)

\int_{\frac{π}{4}}^{\frac{π}{2}} 1 d x = \frac{π}{4}

\int_{\frac{π}{4}}^{\frac{π}{2}} cos (2 x) d x = - \frac{1}{2}

= \frac{1}{2} (\frac{π}{4} - (- \frac{1}{2}))

Simplify

\frac{1}{2} (\frac{π}{4} - (- \frac{1}{2})) : \frac{π + 2}{8}

= \frac{π + 2}{8}

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What is the integral from pi/4 to pi/2 of sin^2(x) ?
The integral from pi/4 to pi/2 of sin^2(x) is (pi+2)/8