Solve ∫ _{-1/2}^frac{1{2}}sin^2(2pif)dt

$$\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \sin ^ { 2 } ( 2 \pi f ) d t$$

Evaluate

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

Solution Steps

Evaluate the indefinite integral first.

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

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

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

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.

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

Simplify.

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

Differentiate w.r.t. f

$$2\pi \sin(4\pi f)$$