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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply Trigonometric Substitution

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

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

Take the constant out:

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

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

Rewrite using trig identities

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

Use the common integral:

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

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

Substitute back

u = arctan (\frac{1}{a} x)

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

Simplify

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

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

Add a constant to the solution

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

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