Solve ∫ 1/4pivarepsilon_{0}cosθdθ

$$\int \frac { 1 } { 4 \pi \varepsilon _ { 0 } } \cos \theta d \theta$$

Evaluate

$$\frac{\sin(\theta )}{4\pi \epsilon _{0}}+С$$

Solution Steps

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

$$\frac{1}{4\pi \epsilon _{0}}\int \cos(\theta )\mathrm{d}\theta$$

Use \int \cos(\epsilon _{0})\mathrm{d}\epsilon _{0}=\sin(\epsilon _{0}) from the table of common integrals to obtain the result.

$$\frac{1}{4\pi \epsilon _{0}}\sin(\theta )$$

Simplify.

$$\frac{\sin(\theta )}{4\pi \epsilon _{0}}$$

If F\left(\theta \right) is an antiderivative of f\left(\theta \right), then the set of all antiderivatives of f\left(\theta \right) is given by F\left(\theta \right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\frac{\sin(\theta )}{4\pi \epsilon _{0}}+С$$

Differentiate w.r.t. ε_0

$$-\frac{\sin(\theta )}{4\pi \epsilon _{0}^{2}}$$