Solve ∫ (tan^2θ+cot^2θ) wrt x=

$$\int ( \tan ^ { 2 } \theta + \cot ^ { 2 } \theta ) d x =$$

Evaluate

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

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

Solution Steps

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

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

Simplify.

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

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.

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

Differentiate w.r.t. x

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