Solve ∫ _{-pi/2}^frac{pi{2}}(1+cos(x)) wrt x

$$\int_{ - \frac{ \pi }{ 2 } }^{ \frac{ \pi }{ 2 } } (1+ \cos ( x ) ) d x$$

Evaluate

$$\pi +2\approx 5.141592654$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

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

$$x+\int \cos(x)\mathrm{d}x$$

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

$$x+\sin(x)$$

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.

$$\frac{\pi }{2}+\sin(\frac{\pi }{2})-\left(-\frac{\pi }{2}+\sin(-\frac{\pi }{2})\right)$$

Simplify.

$$\pi +2$$