Solve ∫ [sec^2(x)-3/2csc^2(x)] wrt y

$$\int [ \sec ^ { 2 } ( x ) - \frac { 3 } { 2 } \csc ^ { 2 } ( x ) ] d y$$

Evaluate

$$\frac{2y\left(2\left(\sin(x)\right)^{2}-3\left(\cos(x)\right)^{2}\right)}{\left(\sin(2x)\right)^{2}}+С$$

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

Solution Steps

Find the integral of \frac{1}{\left(\cos(x)\right)^{2}}-\frac{3}{2\left(\sin(x)\right)^{2}} using the table of common integrals rule \int a\mathrm{d}y=ay.

$$\left(\frac{1}{\left(\cos(x)\right)^{2}}-\frac{3}{2\left(\sin(x)\right)^{2}}\right)y$$

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

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

Differentiate w.r.t. x

$$\frac{8y\left(2\left(\sin(x)\right)^{4}+3\left(\cos(x)\right)^{4}\right)}{\left(\sin(2x)\right)^{3}}$$