Solve ∫ sec^2θ+csc^2θ

$$\int \sec ^ { 2 } \theta + \csc ^ { 2 } \theta$$

Evaluate

$$\frac{4x}{\left(\sin(2\theta )\right)^{2}}+С$$

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

Solution Steps

Find the integral of \frac{1}{\left(\cos(\theta )\right)^{2}}+\frac{1}{\left(\sin(\theta )\right)^{2}} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\left(\frac{1}{\left(\cos(\theta )\right)^{2}}+\frac{1}{\left(\sin(\theta )\right)^{2}}\right)x$$

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

$$\begin{matrix}\left(\frac{1}{\left(\cos(\theta )\right)^{2}}+\frac{1}{\left(\sin(\theta )\right)^{2}}\right)x+С_{3},&\end{matrix}$$

Differentiate w.r.t. x

$$\frac{4}{\left(\sin(2\theta )\right)^{2}}$$