Solve ∫ (from 2 to pi) of 1/2sin(x) wrt x=

$$\int _ { 2 } ^ { \pi } \frac { 1 } { 2 } \sin ( x ) d x =$$

Evaluate

$$\frac{\cos(2)+1}{2}\approx 0.291926582$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{\sin(x)}{2}\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$\frac{\int \sin(x)\mathrm{d}x}{2}$$

Use \int \sin(x)\mathrm{d}x=-\cos(x) from the table of common integrals to obtain the result.

$$-\frac{\cos(x)}{2}$$

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.

$$-\frac{1}{2}\cos(\pi )+\frac{1}{2}\cos(2)$$

Simplify.

$$\frac{1}{2}\left(1+2\times \frac{\cos(2)}{2}\right)$$