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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integral Substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = \frac{x}{a}

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

Simplify

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

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

Add a constant to the solution

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

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