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

x \to \infty lim (x + 6)

\frac{\partial}{\partial x} (\frac{1}{2 x ^{\frac{5}{2} y^{\frac{1}{2}}}})

e^{2 t} sin (5 t) cos (3 t)

x \to 0 + lim (\frac{5}{x ^{2} - x})

x^{2}, x^{3}, 0, 1

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