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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \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)

Simplify

= arctan (x) - x

Add a constant to the solution

= arctan (x) - x + C

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