Solve ∫ (from 0 to frac { pi) of {2}}sin^2t wrt x=

$$\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { 2 } t d x =$$

Evaluate

$$\frac{\pi \left(\sin(t)\right)^{2}}{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \left(\sin(t)\right)^{2}\mathrm{d}x$$

Find the integral of \left(\sin(t)\right)^{2} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\left(\sin(t)\right)^{2}x$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{1}{2}\left(\sin(t)\right)^{2}\pi +0\left(\sin(t)\right)^{2}$$

Simplify.

$$\frac{\left(\sin(t)\right)^{2}\pi }{2}$$

Differentiate w.r.t. t

$$\frac{\pi \sin(2t)}{2}$$