Solve ∫ _{pi/6}^frac{pi{2}}2sin(x) wrt x

$$\int_{ \frac{ \pi }{ 6 } }^{ \frac{ \pi }{ 2 } } 2 \sin ( x ) d x$$

Evaluate

$$\sqrt{3}\approx 1.732050808$$

Solution Steps

Evaluate the indefinite integral first.

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

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

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

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

$$-2\cos(x)$$

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.

$$-2\cos(\frac{1}{2}\pi )+2\cos(\frac{1}{6}\pi )$$

Simplify.

$$\sqrt{3}$$