What is the integral of (1+tan^2(x))/(tan(x)) ?

The integral of (1+tan^2(x))/(tan(x)) is ln|sin(x)|-ln|cos(x)|+C

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integral of (1+tan^2(x))/(tan(x))

\int \frac{1 + tan ^{2} ( x )}{tan ( x )} d x

ln ∣ sin (x) ∣ - ln ∣ cos (x) ∣ + C

Solution steps

\int \frac{1 + tan ^{2} ( x )}{tan ( x )} d x

Expand

\frac{1 + tan ^{2} ( x )}{tan ( x )} : \frac{1}{tan ( x )} + tan (x)

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

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

\int tan (x) d x = - ln ∣ cos (x) ∣

= ln ∣ sin (x) ∣ - ln ∣ cos (x) ∣

Add a constant to the solution

= ln ∣ sin (x) ∣ - ln ∣ cos (x) ∣ + C

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What is the integral of (1+tan^2(x))/(tan(x)) ?
The integral of (1+tan^2(x))/(tan(x)) is ln|sin(x)|-ln|cos(x)|+C