What is the integral of ((3x^2)/(1+x^2)+tan(2x)) ?

The integral of ((3x^2)/(1+x^2)+tan(2x)) is 3(-arctan(x)+x)-1/2 ln|cos(2x)|+C

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integral of ((3x^2)/(1+x^2)+tan(2x))

\int (\frac{3 x ^{2}}{1 + x ^{2}} + tan (2 x)) d x

3 (- arctan (x) + x) - \frac{1}{2} ln ∣ cos (2 x) ∣ + C

Solution steps

\int \frac{3 x ^{2}}{1 + x ^{2}} + tan (2 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{3 x ^{2}}{1 + x ^{2}} d x + \int tan (2 x) d x

\int \frac{3 x ^{2}}{1 + x ^{2}} d x = 3 (- arctan (x) + x)

\int tan (2 x) d x = - \frac{1}{2} ln ∣ cos (2 x) ∣

= 3 (- arctan (x) + x) - \frac{1}{2} ln ∣ cos (2 x) ∣

Add a constant to the solution

= 3 (- arctan (x) + x) - \frac{1}{2} ln ∣ cos (2 x) ∣ + C

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What is the integral of ((3x^2)/(1+x^2)+tan(2x)) ?
The integral of ((3x^2)/(1+x^2)+tan(2x)) is 3(-arctan(x)+x)-1/2 ln|cos(2x)|+C