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

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

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

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

3 (- arctan (x) + x) + C

Solution steps

\int \frac{3 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

= 3 \cdot \int \frac{x ^{2}}{x ^{2} + 1} d x

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

= 3 \cdot \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

= 3 (- \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

= 3 (- arctan (x) + x)

Add a constant to the solution

= 3 (- arctan (x) + x) + C

\frac{d y}{d x} - \frac{1}{x} y = 4

\frac{d}{d t} (e^{- t^{2}})

f (t) = t^{9} sin (t)

\int_{1}^{e} 4 x^{3} ln (x) d x

- \frac{1}{2} x

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