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

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

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

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

- arctan (x) + x + C

Solution steps

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

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

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

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

\int 1 d x = x

= - arctan (x) + x

Add a constant to the solution

= - arctan (x) + x + C

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

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

\int \frac{e ^{7 x}}{e ^{7 x} + 2} d x

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