Solve ∫ cos2θ/cosθdt

$$\int \frac { \cos 2 \theta } { \cos \theta } d t$$

Evaluate

$$\frac{t\cos(2\theta )}{\cos(\theta )}+С$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}$$

Solution Steps

Find the integral of \frac{\cos(2\theta )}{\cos(\theta )} using the table of common integrals rule \int a\mathrm{d}t=at.

$$\frac{\cos(2\theta )}{\cos(\theta )}t$$

Simplify.

$$\frac{\cos(2\theta )t}{\cos(\theta )}$$

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

$$\begin{matrix}\frac{\cos(2\theta )t}{\cos(\theta )}+С_{3},&\end{matrix}$$

Differentiate w.r.t. θ

$$t\left(\frac{\cos(2\theta )\tan(\theta )}{\cos(\theta )}-4\sin(\theta )\right)$$