What is the integral from pi/3 to x of sin^2(t) ?

The integral from pi/3 to x of sin^2(t) is 1/2 (x-pi/3-1/2 (sin(2x)-(sqrt(3))/2))

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integral from pi/3 to x of sin^2(t)

\int_{\frac{π}{3}}^{x} sin^{2} (t) d t

\frac{1}{2} (x - \frac{π}{3} - \frac{1}{2} (sin (2 x) - \frac{3}{2}))

Solution steps

\int_{\frac{π}{3}}^{x} sin^{2} (t) d t

Rewrite using trig identities

= \int_{\frac{π}{3}}^{x} \frac{1 - cos ( 2 t )}{2} d t

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}{3}}^{x} 1 - cos (2 t) d t

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}{3}}^{x} 1 d t - \int_{\frac{π}{3}}^{x} cos (2 t) d t)

\int_{\frac{π}{3}}^{x} 1 d t = x - \frac{π}{3}

\int_{\frac{π}{3}}^{x} cos (2 t) d t = \frac{1}{2} (sin (2 x) - \frac{3}{2})

= \frac{1}{2} (x - \frac{π}{3} - \frac{1}{2} (sin (2 x) - \frac{3}{2}))

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What is the integral from pi/3 to x of sin^2(t) ?
The integral from pi/3 to x of sin^2(t) is 1/2 (x-pi/3-1/2 (sin(2x)-(sqrt(3))/2))