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

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

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

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

\frac{2}{3} a^{2} x^{3} + a + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

Take the constant out:

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

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

Apply u-substitution

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

Apply the Power Rule

= a^{2} \frac{1}{3} \cdot 2 v^{\frac{1}{2}}

Substitute back

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

Simplify

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

= \frac{2}{3} a^{2} x^{3} + a

Add a constant to the solution

= \frac{2}{3} a^{2} x^{3} + a + C

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