Solve ∫ (from 0 to pi) of -20sin(t)-2cos(t)

$$\int _ { 0 } ^ { \pi } - 20 \sin ( t ) - 2 \cos ( t )$$

Evaluate

$$-2\pi \left(\cos(t)+10\sin(t)\right)$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 2\left(-10\sin(t)-\cos(t)\right)\mathrm{d}x$$

Find the integral of 2\left(-10\sin(t)-\cos(t)\right) using the table of common integrals rule \int a\mathrm{d}x=ax.

$$2\left(-10\sin(t)-\cos(t)\right)x$$

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.

$$\left(-20\sin(t)-2\cos(t)\right)\pi +0\left(-20\sin(t)-2\cos(t)\right)$$

Simplify.

$$-2\pi \left(10\sin(t)+\cos(t)\right)$$

Differentiate w.r.t. t

$$2\pi \left(\sin(t)-10\cos(t)\right)$$