Solve ∫ sin^2A+tan^2A/sec^2A

$$\int \frac { \sin ^ { 2 } A + \tan ^ { 2 } A } { \sec ^ { 2 } A }$$

Evaluate

$$x\left(\sin(A)\right)^{2}\left(\left(\cos(A)\right)^{2}+1\right)+С$$

Solution Steps

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

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

Simplify.

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

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.

$$\left(\sin(A)\right)^{2}\left(\left(\cos(A)\right)^{2}+1\right)x+С$$

Differentiate w.r.t. x

$$\left(\sin(A)\right)^{2}+\left(\frac{1}{2}\sin(2A)\right)^{2}$$