What is the integral of (5x^3+5x+11)/((x^2+1)^2) ?

The integral of (5x^3+5x+11)/((x^2+1)^2) is 5/2 ln|x^2+1|+11/2 (arctan(x)+x/(1+x^2))+C

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integral of (5x^3+5x+11)/((x^2+1)^2)

\int \frac{5 x ^{3} + 5 x + 11}{( x ^{2} + 1 ) ^{2}} d x

\frac{5}{2} ln x^{2} + 1 + \frac{11}{2} (arctan (x) + \frac{x}{1 + x ^{2}}) + C

Solution steps

\int \frac{5 x ^{3} + 5 x + 11}{( x ^{2} + 1 ) ^{2}} d x

Take the partial fraction of

\frac{5 x ^{3} + 5 x + 11}{( x ^{2} + 1 ) ^{2}} : \frac{5 x}{x ^{2} + 1} + \frac{11}{( x ^{2} + 1 ) ^{2}}

= \int \frac{5 x}{x ^{2} + 1} + \frac{11}{( x ^{2} + 1 ) ^{2}} 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 \frac{5 x}{x ^{2} + 1} d x + \int \frac{11}{( x ^{2} + 1 ) ^{2}} d x

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

\int \frac{11}{( x ^{2} + 1 ) ^{2}} d x = \frac{11}{2} (arctan (x) + \frac{x}{1 + x ^{2}})

= \frac{5}{2} ln x^{2} + 1 + \frac{11}{2} (arctan (x) + \frac{x}{1 + x ^{2}})

Add a constant to the solution

= \frac{5}{2} ln x^{2} + 1 + \frac{11}{2} (arctan (x) + \frac{x}{1 + x ^{2}}) + C

y^{^{'}} - \frac{y}{x} = x^{2}

\frac{2 s + 5}{s ^{2} - 4 s - 5}

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

\int sin (x) cos (6 x) d x

\frac{d}{d x} ((\frac{x ^{3} - 1}{x ^{3} + 1})^{7})

What is the integral of (5x^3+5x+11)/((x^2+1)^2) ?
The integral of (5x^3+5x+11)/((x^2+1)^2) is 5/2 ln|x^2+1|+11/2 (arctan(x)+x/(1+x^2))+C