integral of (3x)^2 — step-by-step solution

Before integrating, let's calculate:

{(3 x)}^{2}

ⓘ The power of a product is the product of the powers:

{(3 x)}^{2}

3^{2} \cdot x^{2}

3^{2} \cdot x^{2}

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

3^{2}

3 \cdot 3

9

9 x^{2}

Let's compute the integral:

\int 9 x^{2} d x

ⓘ A constant factor stays in front of the integral. An antiderivative of x^n is x^(n+1)/(n+1), for n ≠ -1:

\int 9 x^{2} d x

9 \cdot \frac{x^{3}}{3}

9 \cdot \frac{x^{3}}{3}

ⓘ Multiplying a fraction by a number multiplies its numerator:

9 \cdot \frac{x^{3}}{3}

\frac{9 x^{3}}{3}

■A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{9 x^{3}}{3}

3 x^{3}

3 x^{3}

▶Indefinite integrals are defined up to an additive constant C, so this yields:

3 x^{3} + C