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

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

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

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

3 x + ln x^{2} + 1 + C

Solution steps

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

Expand

\frac{3 x ^{2} + 2 x + 3}{x ^{2} + 1} : \frac{3 x ^{2}}{x ^{2} + 1} + \frac{2 x}{x ^{2} + 1} + \frac{3}{x ^{2} + 1}

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

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

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

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

= 3 (- arctan (x) + x) + ln x^{2} + 1 + 3 arctan (x)

Simplify

3 (- arctan (x) + x) + ln x^{2} + 1 + 3 arctan (x) : 3 x + ln x^{2} + 1

= 3 x + ln x^{2} + 1

Add a constant to the solution

= 3 x + ln x^{2} + 1 + C

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