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

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

Popular >

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

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

2 ln ∣ sin (x) ∣ - ln ∣ cos (x) ∣ + C

Solution steps

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

Expand

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

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

\int \frac{2}{tan ( x )} d x = 2 ln ∣ sin (x) ∣

\int tan (x) d x = - ln ∣ cos (x) ∣

= 2 ln ∣ sin (x) ∣ - ln ∣ cos (x) ∣

Add a constant to the solution

= 2 ln ∣ sin (x) ∣ - ln ∣ cos (x) ∣ + C

\int \frac{x + 2}{( x - 1 ) ^{2} ( x ^{2} + 4 )} d x

\int \frac{( x + 1 ) ^{6}}{2 x} d x

\int \frac{1}{5 x + x ( ln ( x ) ) ^{2}} d x

\int \frac{1}{2 x + 1} d x

\int \frac{x}{( x + a ) ( x - a )} d x

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