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

The integral of (x^3)/((x^2+a^2)^3) is (-2x^2-a^2)/(4(x^2+a^2)^2)+C

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

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

\frac{- 2 x ^{2} - a ^{2}}{4 ( x ^{2} + a ^{2} ) ^{2}} + C

Solution steps

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

Apply Integral Substitution

= \int \frac{a ^{4} u ^{3}}{( a ^{2} u ^{2} + a ^{2} ) ^{3}} d u

Take the constant out:

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

= a^{4} \cdot \int \frac{u ^{3}}{( a ^{2} u ^{2} + a ^{2} ) ^{3}} d u

Apply u-substitution

: \int \frac{v - a ^{2}}{2 a ^{4} v ^{3}} d v

= a^{4} \cdot \int \frac{v - a ^{2}}{2 a ^{4} v ^{3}} d v

Take the constant out:

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

= a^{4} \frac{1}{2 a ^{4}} \cdot \int \frac{v - a ^{2}}{v ^{3}} d v

Expand

\frac{v - a ^{2}}{v ^{3}} : \frac{1}{v ^{2}} - \frac{a ^{2}}{v ^{3}}

= a^{4} \frac{1}{2 a ^{4}} \cdot \int \frac{1}{v ^{2}} - \frac{a ^{2}}{v ^{3}} d v

Apply the Sum Rule:

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

= a^{4} \frac{1}{2 a ^{4}} (\int \frac{1}{v ^{2}} d v - \int \frac{a ^{2}}{v ^{3}} d v)

\int \frac{1}{v ^{2}} d v = - \frac{1}{v}

\int \frac{a ^{2}}{v ^{3}} d v = - \frac{a ^{2}}{2 v ^{2}}

= a^{4} \frac{1}{2 a ^{4}} (- \frac{1}{v} - (- \frac{a ^{2}}{2 v ^{2}}))

Substitute back

= a^{4} \frac{1}{2 a ^{4}} - \frac{1}{a ^{2} ( \frac{x}{a} ) ^{2} + a ^{2}} - - \frac{a ^{2}}{2 ( a ^{2} ( \frac{x}{a} ) ^{2} + a ^{2} ) ^{2}}

Simplify

a^{4} \frac{1}{2 a ^{4}} - \frac{1}{a ^{2} ( \frac{x}{a} ) ^{2} + a ^{2}} - - \frac{a ^{2}}{2 ( a ^{2} ( \frac{x}{a} ) ^{2} + a ^{2} ) ^{2}} : \frac{- 2 x ^{2} - a ^{2}}{4 ( x ^{2} + a ^{2} ) ^{2}}

= \frac{- 2 x ^{2} - a ^{2}}{4 ( x ^{2} + a ^{2} ) ^{2}}

Add a constant to the solution

= \frac{- 2 x ^{2} - a ^{2}}{4 ( x ^{2} + a ^{2} ) ^{2}} + C

y = 3 x, x + y = 4, y = 1

\int_{- 1}^{1} \frac{5 r}{( 4 + r ^{2} ) ^{2}} d r

\frac{\partial}{\partial x} (\frac{x}{x ^{2} + x y + y ^{2}})

\int \frac{3 x - 2}{5 x ^{2} - 3 x + 2} d x

55 cos (0.03 x - 1.11) + 86

What is the integral of (x^3)/((x^2+a^2)^3) ?
The integral of (x^3)/((x^2+a^2)^3) is (-2x^2-a^2)/(4(x^2+a^2)^2)+C