What is the integral of (2x^2+5)/(x^2+1) ?

The integral of (2x^2+5)/(x^2+1) is 2x+3arctan(x)+C

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integral of (2x^2+5)/(x^2+1)

\int \frac{2 x ^{2} + 5}{x ^{2} + 1} d x

2 x + 3 arctan (x) + C

Solution steps

\int \frac{2 x ^{2} + 5}{x ^{2} + 1} d x

Long division

\frac{2 x ^{2} + 5}{x ^{2} + 1} : 2 + \frac{3}{x ^{2} + 1}

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

\int 2 d x = 2 x

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

= 2 x + 3 arctan (x)

Add a constant to the solution

= 2 x + 3 arctan (x) + C

\frac{d}{d y} (\frac{z}{x ^{2} + y ^{2} + z ^{2}})

e^{3 t^{2}}

\int_{0}^{3} \frac{1}{1 + x ^{2}} d x

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

\frac{d}{d x} (\frac{x ^{1 - a}}{1 - a})

What is the integral of (2x^2+5)/(x^2+1) ?
The integral of (2x^2+5)/(x^2+1) is 2x+3arctan(x)+C