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

The integral of (x^3)/((x^2+a^2)^{3/2)} is (x^2+2a^2)/((x^2+a^2)^{1/2)}+C

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

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

\frac{x ^{2} + 2 a ^{2}}{( x ^{2} + a ^{2} ) ^{\frac{1}{2}}} + C

Solution steps

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

Apply u-substitution

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

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

Apply u-substitution

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

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

Expand

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

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

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

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

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

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of (x^3)/((x^2+a^2)^{3/2)} ?
The integral of (x^3)/((x^2+a^2)^{3/2)} is (x^2+2a^2)/((x^2+a^2)^{1/2)}+C