Solve ∫ (from 0 to frac { pi) of {3}}(cosx-1/2) wrt x

$$\int _ { 0 } ^ { \frac { \pi } { 3 } } ( \cos x - \frac { 1 } { 2 } ) d x$$

Evaluate

$$\frac{\sqrt{3}}{2}-\frac{\pi }{6}\approx 0.342426628$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{1}{2}\left(2\cos(x)-1\right)\mathrm{d}x$$

Integrate the sum term by term.

$$\int \cos(x)\mathrm{d}x+\int -\frac{1}{2}\mathrm{d}x$$

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

$$\sin(x)+\int -\frac{1}{2}\mathrm{d}x$$

Find the integral of -\frac{1}{2} using the table of common integrals rule \int a\mathrm{d}x=ax.

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

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.

$$\sin(\frac{\pi }{3})-\frac{1}{2}\times \frac{1}{3}\pi -\left(\sin(0)-\frac{1}{2}\times 0\right)$$

Simplify.

$$\frac{\sqrt{3}}{2}-\frac{\pi }{6}$$