What is the integral of 1/(sqrt((x^2+a^2)^3)) ?

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

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integral of 1/(sqrt((x^2+a^2)^3))

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

\frac{x}{a ^{2} a ^{2} + x ^{2}} + C

Solution steps

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

Apply Integral Substitution

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

Apply Trigonometric Substitution

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

Use the common integral:

\int cos (v) d v = sin (v)

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

Substitute back

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

Simplify

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

= \frac{x}{a ^{2} a ^{2} + x ^{2}}

Add a constant to the solution

= \frac{x}{a ^{2} a ^{2} + x ^{2}} + C

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