Solve ∫ (from pi / a to pi) of sinθdθ

$$\int _ { \pi / a } ^ { \pi } \sin \theta d \theta$$

Evaluate

$$-\left(\lim_{\theta \to \frac{\pi }{a}}-\cos(\theta )\right)+1$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \sin(\theta )\mathrm{d}\theta$$

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

$$-\cos(\theta )$$

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.

$$-\cos(\pi )+\cos(\pi a^{-1})$$

Simplify.

$$\left\{\begin{matrix}1+\cos(\frac{\pi }{a}),&\\\text{Indeterminate},&a=0\end{matrix}\right.$$

Differentiate w.r.t. a

$$-\frac{\mathrm{d}}{\mathrm{d}a}(\lim_{\theta \to \frac{\pi }{a}}-\cos(\theta ))$$