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

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

Popular >

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

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

2 x - 3 arctan (x) + C

Solution steps

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

Expand

\frac{2 x ^{2} - 1}{1 + x ^{2}} : \frac{2 x ^{2}}{1 + x ^{2}} - \frac{1}{1 + x ^{2}}

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

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

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

= 2 (- arctan (x) + x) - arctan (x)

Simplify

2 (- arctan (x) + x) - arctan (x) : 2 x - 3 arctan (x)

= 2 x - 3 arctan (x)

Add a constant to the solution

= 2 x - 3 arctan (x) + C

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

\int \frac{x ^{2} - 5}{x ^{2}} d x

\int (sec (2 θ) - csc (\frac{θ}{2})) d θ

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

\int 2 sin (2 θ) sin (4 θ) d θ

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