What is the integral of 1/((u^2+a^2)^2) ?

The integral of 1/((u^2+a^2)^2) is 1/(2a^3)(arctan(u/a)+1/2 sin(2arctan(u/a)))+C

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integral of 1/((u^2+a^2)^2)

\int \frac{1}{( u ^{2} + a ^{2} ) ^{2}} d u

\frac{1}{2 a ^{3}} (arctan (\frac{u}{a}) + \frac{1}{2} sin (2 arctan (\frac{u}{a}))) + C

Solution steps

\int \frac{1}{( u ^{2} + a ^{2} ) ^{2}} d u

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

= \int \frac{1}{( a ^{2} ( 1 + \frac{u ^{2}}{a ^{2}} ) ) ^{2}} d u

Apply Integral Substitution

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

Take the constant out:

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

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

Apply Trigonometric Substitution

= \frac{1}{a ^{3}} \cdot \int \frac{1}{sec ^{4} ( w ) cos ^{2} ( w )} d w

Rewrite using trig identities

= \frac{1}{a ^{3}} \cdot \int \frac{1 + cos ( 2 w )}{2} d w

Take the constant out:

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

= \frac{1}{a ^{3}} \cdot \frac{1}{2} \cdot \int 1 + cos (2 w) d w

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}{a ^{3}} \cdot \frac{1}{2} (\int 1 d w + \int cos (2 w) d w)

\int 1 d w = w

\int cos (2 w) d w = \frac{1}{2} sin (2 w)

= \frac{1}{a ^{3}} \cdot \frac{1}{2} (w + \frac{1}{2} sin (2 w))

Substitute back

= \frac{1}{a ^{3}} \cdot \frac{1}{2} (arctan (\frac{u}{a}) + \frac{1}{2} sin (2 arctan (\frac{u}{a})))

Simplify

\frac{1}{a ^{3}} \cdot \frac{1}{2} (arctan (\frac{u}{a}) + \frac{1}{2} sin (2 arctan (\frac{u}{a}))) : \frac{1}{2 a ^{3}} (arctan (\frac{u}{a}) + \frac{1}{2} sin (2 arctan (\frac{u}{a})))

= \frac{1}{2 a ^{3}} (arctan (\frac{u}{a}) + \frac{1}{2} sin (2 arctan (\frac{u}{a})))

Add a constant to the solution

= \frac{1}{2 a ^{3}} (arctan (\frac{u}{a}) + \frac{1}{2} sin (2 arctan (\frac{u}{a}))) + C

What is the integral of 1/((u^2+a^2)^2) ?
The integral of 1/((u^2+a^2)^2) is 1/(2a^3)(arctan(u/a)+1/2 sin(2arctan(u/a)))+C