Solve ∫ _{-pi/2}^frac{pi{2}}x wrt x

$$\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } x d x$$

Evaluate

$$0$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}.

$$\frac{x^{2}}{2}$$

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.

$$\frac{1}{2}\times \left(\frac{1}{2}\pi \right)^{2}-\frac{1}{2}\left(-\frac{1}{2}\pi \right)^{2}$$

Simplify.

$$0$$