Solve ∫ (from pi / 4 to frac { 5 pi) of { 2}}-8sinx wrt x

$$\int _ { \pi / 4 } ^ { \frac { 5 \pi } { 2 } } - 8 \sin x d x$$

Evaluate

$$-4\sqrt{2}\approx -5.656854249$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -8\sin(x)\mathrm{d}x$$

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

$$-8\int \sin(x)\mathrm{d}x$$

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

$$8\cos(x)$$

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.

$$8\cos(\frac{1}{2}\times 5\pi )-8\cos(\frac{1}{4}\pi )$$

Simplify.

$$-4\sqrt{2}$$