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

The integral of [(a^3)/(x^2+a^2)]^2 is (a^3)/4 (2arctan(x/a)+sin(2arctan(x/a)))+C

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

\int [\frac{a ^{3}}{x ^{2} + a ^{2}}]^{2} d x

\frac{a ^{3}}{4} (2 arctan (\frac{x}{a}) + sin (2 arctan (\frac{x}{a}))) + C

Solution steps

\int [\frac{a ^{3}}{x ^{2} + a ^{2}}]^{2} d x

Simplify

[\frac{a ^{3}}{x ^{2} + a ^{2}}]^{2} : \frac{a ^{6}}{( x ^{2} + a ^{2} ) ^{2}}

= \int \frac{a ^{6}}{( x ^{2} + a ^{2} ) ^{2}} d x

Take the constant out:

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

= a^{6} \cdot \int \frac{1}{( x ^{2} + a ^{2} ) ^{2}} d x

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

= a^{6} \cdot \int \frac{1}{( a ^{2} ( \frac{x ^{2}}{a ^{2}} + 1 ) ) ^{2}} d x

Apply Integral Substitution

= a^{6} \cdot \int \frac{1}{a ^{3} ( u ^{2} + 1 ) ^{2}} d u

Take the constant out:

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

= a^{6} \frac{1}{a ^{3}} \cdot \int \frac{1}{( u ^{2} + 1 ) ^{2}} d u

Apply Trigonometric Substitution

= a^{6} \frac{1}{a ^{3}} \cdot \int \frac{1}{sec ^{4} ( v ) cos ^{2} ( v )} d v

Rewrite using trig identities

= a^{6} \frac{1}{a ^{3}} \cdot \int \frac{1 + cos ( 2 v )}{2} d v

Take the constant out:

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

= a^{6} \frac{1}{a ^{3}} \cdot \frac{1}{2} \cdot \int 1 + cos (2 v) 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^{6} \frac{1}{a ^{3}} \cdot \frac{1}{2} (\int 1 d v + \int cos (2 v) d v)

\int 1 d v = v

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

= a^{6} \frac{1}{a ^{3}} \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Substitute back

= a^{6} \frac{1}{a ^{3}} \cdot \frac{1}{2} (arctan (\frac{x}{a}) + \frac{1}{2} sin (2 arctan (\frac{x}{a})))

Simplify

a^{6} \frac{1}{a ^{3}} \cdot \frac{1}{2} (arctan (\frac{x}{a}) + \frac{1}{2} sin (2 arctan (\frac{x}{a}))) : \frac{a ^{3}}{4} (2 arctan (\frac{x}{a}) + sin (2 arctan (\frac{x}{a})))

= \frac{a ^{3}}{4} (2 arctan (\frac{x}{a}) + sin (2 arctan (\frac{x}{a})))

Add a constant to the solution

= \frac{a ^{3}}{4} (2 arctan (\frac{x}{a}) + sin (2 arctan (\frac{x}{a}))) + C

What is the integral of [(a^3)/(x^2+a^2)]^2 ?
The integral of [(a^3)/(x^2+a^2)]^2 is (a^3)/4 (2arctan(x/a)+sin(2arctan(x/a)))+C