Solve ∫ (from pi / 4 to 5 pi / 2) of -4sin(x) wrt x

$$\int _ { \pi / 4 } ^ { 5 \pi / 2 } - 4 \sin ( x ) d x$$

Evaluate

$$-2\sqrt{2}\approx -2.828427125$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -4\sin(x)\mathrm{d}x$$

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

$$-4\int \sin(x)\mathrm{d}x$$

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

$$4\cos(x)$$

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.

$$4\cos(\frac{1}{2}\times 5\pi )-4\cos(\frac{1}{4}\pi )$$

Simplify.

$$-2\sqrt{2}$$