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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

x^{2} = (x),

assuming

x \geq 0

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

Cancel the common factor:

x

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

Apply the Power Rule

: \frac{x ^{3}}{3}

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

Simplify

3 a^{2} \frac{x ^{3}}{3} : a^{2} x^{3}

= a^{2} x^{3}

Add a constant to the solution

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

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