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

The integral of ((3x^2+5))/((x^2+1)^2) is -(3x)/(2(x^2+1))+4arctan(x)+5/2*x/(x^2+1)+C

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integral of ((3x^2+5))/((x^2+1)^2)

integral

\frac{( 3 x ^{2} + 5 )}{( x ^{2} + 1 ) ^{2}}

- \frac{3 x}{2 ( x ^{2} + 1 )} + 4 arctan (x) + \frac{5}{2} \cdot \frac{x}{x ^{2} + 1} + C

Solution steps

\int \frac{3 x ^{2} + 5}{( x ^{2} + 1 ) ^{2}} d x

Expand

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

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

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

\int \frac{5}{( x ^{2} + 1 ) ^{2}} d x = \frac{5}{2} (arctan (x) + \frac{x}{1 + x ^{2}})

= 3 (\frac{1}{2} arctan (x) - \frac{x}{2 ( 1 + x ^{2} )}) + \frac{5}{2} (arctan (x) + \frac{x}{1 + x ^{2}})

Simplify

3 (\frac{1}{2} arctan (x) - \frac{x}{2 ( 1 + x ^{2} )}) + \frac{5}{2} (arctan (x) + \frac{x}{1 + x ^{2}}) : - \frac{3 x}{2 ( x ^{2} + 1 )} + 4 arctan (x) + \frac{5}{2} \cdot \frac{x}{x ^{2} + 1}

= - \frac{3 x}{2 ( x ^{2} + 1 )} + 4 arctan (x) + \frac{5}{2} \cdot \frac{x}{x ^{2} + 1}

Add a constant to the solution

= - \frac{3 x}{2 ( x ^{2} + 1 )} + 4 arctan (x) + \frac{5}{2} \cdot \frac{x}{x ^{2} + 1} + C

2 - (\frac{5}{x ^{2}})

1 - x^{2}

∣ x ∣^{4}

\frac{7}{x}

(x^{2}) \cdot e^{x^{3}}

What is the integral of ((3x^2+5))/((x^2+1)^2) ?
The integral of ((3x^2+5))/((x^2+1)^2) is -(3x)/(2(x^2+1))+4arctan(x)+5/2*x/(x^2+1)+C