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

The integral of 1/(x(x^2+3)^2) is 1/9 ln(x)-1/18 ln|x^2+3|+1/(6(x^2+3))+C

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

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

\frac{1}{9} ln (x) - \frac{1}{18} ln x^{2} + 3 + \frac{1}{6 ( x ^{2} + 3 )} + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{2 ( u - 3 ) u ^{2}} d u

= \int \frac{1}{2 ( u - 3 ) u ^{2}} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{( u - 3 ) u ^{2}} d u

Take the partial fraction of

\frac{1}{( u - 3 ) u ^{2}} : \frac{1}{9 ( u - 3 )} - \frac{1}{9 u} - \frac{1}{3 u ^{2}}

= \frac{1}{2} \cdot \int \frac{1}{9 ( u - 3 )} - \frac{1}{9 u} - \frac{1}{3 u ^{2}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int \frac{1}{9 ( u - 3 )} d u - \int \frac{1}{9 u} d u - \int \frac{1}{3 u ^{2}} d u)

\int \frac{1}{9 ( u - 3 )} d u = \frac{1}{9} ln ∣ u - 3 ∣

\int \frac{1}{9 u} d u = \frac{1}{9} ln ∣ u ∣

\int \frac{1}{3 u ^{2}} d u = - \frac{1}{3 u}

= \frac{1}{2} (\frac{1}{9} ln ∣ u - 3 ∣ - \frac{1}{9} ln ∣ u ∣ - (- \frac{1}{3 u}))

Substitute back

u = x^{2} + 3

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

Simplify

\frac{1}{2} (\frac{1}{9} ln x^{2} + 3 - 3 - \frac{1}{9} ln x^{2} + 3 - (- \frac{1}{3 ( x ^{2} + 3 )})) : \frac{1}{9} ln (x) - \frac{1}{18} ln x^{2} + 3 + \frac{1}{6 ( x ^{2} + 3 )}

= \frac{1}{9} ln (x) - \frac{1}{18} ln x^{2} + 3 + \frac{1}{6 ( x ^{2} + 3 )}

Add a constant to the solution

= \frac{1}{9} ln (x) - \frac{1}{18} ln x^{2} + 3 + \frac{1}{6 ( x ^{2} + 3 )} + C

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What is the integral of 1/(x(x^2+3)^2) ?
The integral of 1/(x(x^2+3)^2) is 1/9 ln(x)-1/18 ln|x^2+3|+1/(6(x^2+3))+C