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

The integral of 1/(1+2tan(x)) is 1/5 (-ln(sec^2(x))+arctan(tan(x)))+2/5 ln|2tan(x)+1|+C

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

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

\frac{1}{5} (- ln (sec^{2} (x)) + arctan (tan (x))) + \frac{2}{5} ln ∣ 2 tan (x) + 1 ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{( 1 + u ^{2} ) ( 1 + 2 u )} d u

= \int \frac{1}{( 1 + u ^{2} ) ( 1 + 2 u )} d u

Take the partial fraction of

\frac{1}{( 1 + u ^{2} ) ( 1 + 2 u )} : \frac{- 2 u + 1}{5 ( u ^{2} + 1 )} + \frac{4}{5 ( 2 u + 1 )}

= \int \frac{- 2 u + 1}{5 ( u ^{2} + 1 )} + \frac{4}{5 ( 2 u + 1 )} d u

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 u + 1}{5 ( u ^{2} + 1 )} d u + \int \frac{4}{5 ( 2 u + 1 )} d u

\int \frac{- 2 u + 1}{5 ( u ^{2} + 1 )} d u = \frac{1}{5} (- ln u^{2} + 1 + arctan (u))

\int \frac{4}{5 ( 2 u + 1 )} d u = \frac{2}{5} ln ∣ 2 u + 1 ∣

= \frac{1}{5} (- ln u^{2} + 1 + arctan (u)) + \frac{2}{5} ln ∣ 2 u + 1 ∣

Substitute back

u = tan (x)

= \frac{1}{5} (- ln tan^{2} (x) + 1 + arctan (tan (x))) + \frac{2}{5} ln ∣ 2 tan (x) + 1 ∣

ln tan^{2} (x) + 1 = ln (sec^{2} (x))

= \frac{1}{5} (- ln (sec^{2} (x)) + arctan (tan (x))) + \frac{2}{5} ln ∣ 2 tan (x) + 1 ∣

Add a constant to the solution

= \frac{1}{5} (- ln (sec^{2} (x)) + arctan (tan (x))) + \frac{2}{5} ln ∣ 2 tan (x) + 1 ∣ + C

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What is the integral of 1/(1+2tan(x)) ?
The integral of 1/(1+2tan(x)) is 1/5 (-ln(sec^2(x))+arctan(tan(x)))+2/5 ln|2tan(x)+1|+C