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

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

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

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

\frac{π - 2}{4}

Decimal

0.28539 \dots

Solution steps

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

Rewrite using trig identities

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

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

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

= \frac{1}{2} (\frac{π}{2} - 1)

Simplify

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

= \frac{π - 2}{4}

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