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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

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}}^{x^{2}} 1 - cos (2 u) d u

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

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

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

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

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