Solve ∫ (from 0 to frac{ pi) of { 3}}6sin(x) wrt x

$$\int_{ 0 }^{ \frac{ \pi }{ 3 } } 6 \sin ( x ) d x$$

Evaluate

$$3$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 6\sin(x)\mathrm{d}x$$

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

$$6\int \sin(x)\mathrm{d}x$$

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

$$-6\cos(x)$$

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.

$$-6\cos(\frac{1}{3}\pi )+6\cos(0)$$

Simplify.

$$3$$