What is the integral of (x^3)/((x^2+12)^3) ?

The integral of (x^3)/((x^2+12)^3) is -1/(2(x^2+12))+3/((x^2+12)^2)+C

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integral of (x^3)/((x^2+12)^3)

\int \frac{x ^{3}}{( x ^{2} + 12 ) ^{3}} d x

- \frac{1}{2 ( x ^{2} + 12 )} + \frac{3}{( x ^{2} + 12 ) ^{2}} + C

Solution steps

\int \frac{x ^{3}}{( x ^{2} + 12 ) ^{3}} d x

Take the partial fraction of

\frac{x ^{3}}{( x ^{2} + 12 ) ^{3}} : \frac{x}{( x ^{2} + 12 ) ^{2}} - \frac{12 x}{( x ^{2} + 12 ) ^{3}}

= \int \frac{x}{( x ^{2} + 12 ) ^{2}} - \frac{12 x}{( x ^{2} + 12 ) ^{3}} 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{x}{( x ^{2} + 12 ) ^{2}} d x - \int \frac{12 x}{( x ^{2} + 12 ) ^{3}} d x

\int \frac{x}{( x ^{2} + 12 ) ^{2}} d x = - \frac{1}{2 ( x ^{2} + 12 )}

\int \frac{12 x}{( x ^{2} + 12 ) ^{3}} d x = - \frac{3}{( x ^{2} + 12 ) ^{2}}

= - \frac{1}{2 ( x ^{2} + 12 )} - (- \frac{3}{( x ^{2} + 12 ) ^{2}})

Simplify

= - \frac{1}{2 ( x ^{2} + 12 )} + \frac{3}{( x ^{2} + 12 ) ^{2}}

Add a constant to the solution

= - \frac{1}{2 ( x ^{2} + 12 )} + \frac{3}{( x ^{2} + 12 ) ^{2}} + C

\frac{d y}{d x} = \frac{1}{x ^{2} y ^{2}}

x^{2} e^{4 x}

\frac{d y}{d x} = e^{- ka \cdot x}, y (0) = 0

\int \frac{x ^{2} - x + 12}{x ^{3} + 6 x} d x

\frac{\partial}{\partial x} (e^{3 x} sin (cy))

What is the integral of (x^3)/((x^2+12)^3) ?
The integral of (x^3)/((x^2+12)^3) is -1/(2(x^2+12))+3/((x^2+12)^2)+C