Solve ∫ intsinxcosxtanx wrt xdy

$$\int \int \sin x \cos x \tan x d x d y$$

Evaluate

$$\frac{y\left(-\sin(2x)+2x\right)}{4}+Сy+С_{1}$$

Solution Steps

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

$$\int \frac{\cos(x)\left(\sin(x)\right)^{2}}{\cos(x)}\mathrm{d}xy$$

Simplify.

$$\left(\frac{x}{2}-\frac{\sin(2x)}{4}+С\right)y$$

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.

$$\left(\frac{x}{2}-\frac{\sin(2x)}{4}+С\right)y+С$$

Differentiate w.r.t. x

$$y\left(\sin(x)\right)^{2}$$