Solve ∫ 1+tany/y

$$\int \frac { 1 + \tan y } { y }$$

Evaluate

$$\frac{x\left(\sin(y)+\cos(y)\right)}{y\cos(y)}+С$$

Solution Steps

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

$$\frac{1+\frac{\sin(y)}{\cos(y)}}{y}x$$

Simplify.

$$\frac{\left(1+\frac{\sin(y)}{\cos(y)}\right)x}{y}$$

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.

$$\frac{\left(1+\frac{\sin(y)}{\cos(y)}\right)x}{y}+С$$

Differentiate w.r.t. x

$$\frac{\sin(y)+\cos(y)}{y\cos(y)}$$