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

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

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

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

\frac{2}{3 a} arctan (\frac{1}{a} x) + C

Solution steps

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

Take the constant out:

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

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

Simplify

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

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

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

Take the constant out:

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

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

Apply Integral Substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = \frac{1}{a} x

= \frac{2}{3} \cdot \frac{1}{a} a arctan (\frac{1}{a} x)

Simplify

\frac{2}{3} \cdot \frac{1}{a} a arctan (\frac{1}{a} x) : \frac{2}{3 a} arctan (\frac{1}{a} x)

= \frac{2}{3 a} arctan (\frac{1}{a} x)

Add a constant to the solution

= \frac{2}{3 a} arctan (\frac{1}{a} x) + C

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