What is the integral of (sin(x)+tan(x))/(tan(x)) ?

The integral of (sin(x)+tan(x))/(tan(x)) is sin(x)+x+C

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integral of (sin(x)+tan(x))/(tan(x))

\int \frac{sin ( x ) + tan ( x )}{tan ( x )} d x

sin (x) + x + C

Solution steps

\int \frac{sin ( x ) + tan ( x )}{tan ( x )} d x

Expand

\frac{sin ( x ) + tan ( x )}{tan ( x )} : \frac{sin ( x )}{tan ( x )} + 1

= \int \frac{sin ( x )}{tan ( x )} + 1 d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{sin ( x )}{tan ( x )} d x + \int 1 d x

\int \frac{sin ( x )}{tan ( x )} d x = sin (x)

\int 1 d x = x

= sin (x) + x

Add a constant to the solution

= sin (x) + x + C

\int \frac{3 x ^{2} - 6 x + 7}{x ^{3} - x ^{2} - x + 1} d x

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\int \frac{sinh ( 4 x )}{e ^{3 x}} d x

\int (x^{3} + 2 x^{2} + x - 1) e^{2 x} d x

\int \frac{2 x ^{2} + 35}{x ^{4} - 625} d x

What is the integral of (sin(x)+tan(x))/(tan(x)) ?
The integral of (sin(x)+tan(x))/(tan(x)) is sin(x)+x+C