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

The integral of (u^2)/(a^2+u^2) is -aarctan(u/a)+u+C

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integral of (u^2)/(a^2+u^2)

\int \frac{u ^{2}}{a ^{2} + u ^{2}} d u

- a arctan (\frac{u}{a}) + u + C

Solution steps

\int \frac{u ^{2}}{a ^{2} + u ^{2}} d u

Apply Integral Substitution

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

Take the constant out:

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

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

\frac{v ^{2}}{1 + v ^{2}} = - \frac{1}{1 + v ^{2}} + 1

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

\int \frac{1}{1 + v ^{2}} d v = arctan (v)

\int 1 d v = v

= a (- arctan (v) + v)

Substitute back

v = \frac{u}{a}

= a (- arctan (\frac{u}{a}) + \frac{u}{a})

Simplify

a (- arctan (\frac{u}{a}) + \frac{u}{a}) : - a arctan (\frac{u}{a}) + u

= - a arctan (\frac{u}{a}) + u

Add a constant to the solution

= - a arctan (\frac{u}{a}) + u + C

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What is the integral of (u^2)/(a^2+u^2) ?
The integral of (u^2)/(a^2+u^2) is -aarctan(u/a)+u+C