What is the integral of (10)/((x+1)(x^2+9)^2) ?

The integral of (10)/((x+1)(x^2+9)^2) is

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integral of (10)/((x+1)(x^2+9)^2)

\int \frac{10}{( x + 1 ) ( x ^{2} + 9 ) ^{2}} d x

10 (\frac{1}{100} ln ∣ x + 1 ∣ + \frac{1}{100} (- \frac{1}{2} ln x^{2} + 9 + \frac{1}{3} arctan (\frac{x}{3})) + \frac{- x ^{3} sin ( 2 arctan ( \frac{x}{3} ) ) + 2 x ^{2} arctan ( \frac{x}{3} ) + x ^{2} sin ( 2 arctan ( \frac{x}{3} ) ) + 6 x ^{2} - 9 x sin ( 2 arctan ( \frac{x}{3} ) ) + 18 arctan ( \frac{x}{3} ) + 9 sin ( 2 arctan ( \frac{x}{3} ) ) + 54}{1080 ( x ^{2} + 9 )}) + C

Solution steps

\int \frac{10}{( x + 1 ) ( x ^{2} + 9 ) ^{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 10 \cdot \int \frac{1}{( x + 1 ) ( x ^{2} + 9 ) ^{2}} d x

Take the partial fraction of

\frac{1}{( x + 1 ) ( x ^{2} + 9 ) ^{2}} : \frac{1}{100 ( x + 1 )} + \frac{- x + 1}{100 ( x ^{2} + 9 )} + \frac{- x + 1}{10 ( x ^{2} + 9 ) ^{2}}

= 10 \cdot \int \frac{1}{100 ( x + 1 )} + \frac{- x + 1}{100 ( x ^{2} + 9 )} + \frac{- x + 1}{10 ( x ^{2} + 9 ) ^{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

= 10 (\int \frac{1}{100 ( x + 1 )} d x + \int \frac{- x + 1}{100 ( x ^{2} + 9 )} d x + \int \frac{- x + 1}{10 ( x ^{2} + 9 ) ^{2}} d x)

\int \frac{1}{100 ( x + 1 )} d x = \frac{1}{100} ln ∣ x + 1 ∣

\int \frac{- x + 1}{100 ( x ^{2} + 9 )} d x = \frac{1}{100} (- \frac{1}{2} ln x^{2} + 9 + \frac{1}{3} arctan (\frac{x}{3}))

\int \frac{- x + 1}{10 ( x ^{2} + 9 ) ^{2}} d x = \frac{- x ^{3} sin ( 2 arctan ( \frac{x}{3} ) ) + 2 x ^{2} arctan ( \frac{x}{3} ) + x ^{2} sin ( 2 arctan ( \frac{x}{3} ) ) + 6 x ^{2} - 9 x sin ( 2 arctan ( \frac{x}{3} ) ) + 18 arctan ( \frac{x}{3} ) + 9 sin ( 2 arctan ( \frac{x}{3} ) ) + 54}{1080 ( x ^{2} + 9 )}

= 10 (\frac{1}{100} ln ∣ x + 1 ∣ + \frac{1}{100} (- \frac{1}{2} ln x^{2} + 9 + \frac{1}{3} arctan (\frac{x}{3})) + \frac{- x ^{3} sin ( 2 arctan ( \frac{x}{3} ) ) + 2 x ^{2} arctan ( \frac{x}{3} ) + x ^{2} sin ( 2 arctan ( \frac{x}{3} ) ) + 6 x ^{2} - 9 x sin ( 2 arctan ( \frac{x}{3} ) ) + 18 arctan ( \frac{x}{3} ) + 9 sin ( 2 arctan ( \frac{x}{3} ) ) + 54}{1080 ( x ^{2} + 9 )})

Add a constant to the solution

= 10 (\frac{1}{100} ln ∣ x + 1 ∣ + \frac{1}{100} (- \frac{1}{2} ln x^{2} + 9 + \frac{1}{3} arctan (\frac{x}{3})) + \frac{- x ^{3} sin ( 2 arctan ( \frac{x}{3} ) ) + 2 x ^{2} arctan ( \frac{x}{3} ) + x ^{2} sin ( 2 arctan ( \frac{x}{3} ) ) + 6 x ^{2} - 9 x sin ( 2 arctan ( \frac{x}{3} ) ) + 18 arctan ( \frac{x}{3} ) + 9 sin ( 2 arctan ( \frac{x}{3} ) ) + 54}{1080 ( x ^{2} + 9 )}) + C

x \to 8 - lim (\frac{x - 8}{∣ x - 8 ∣})

\frac{d}{d x} (2 (4 - 9 x)^{4})

(x y + y + y^{2}) d x + (x + 2 y) d y = 0

f (x) = x - 5, at x = 6

x = 7 y^{3}, x = 18 y^{2}

What is the integral of (10)/((x+1)(x^2+9)^2) ?
The integral of (10)/((x+1)(x^2+9)^2) is