integral of sqrt(3x-12) — step-by-step solution

1) Let's compute the integral:

\int \sqrt{3 x - 12} d x

For the math expression:

\sqrt{3 x - 12}

, since (a^n)' = n*a'*a^(n - 1), we try differentiating

{(3 x - 12)}^{\frac{1}{2} + 1}

2) Let's differentiate:

\frac{d}{d x} ({(3 x - 12)}^{\frac{1}{2} + 1})

ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{1}{2} + 1

\frac{1 + 2}{2}

\frac{d}{d x} ({(3 x - 12)}^{\frac{1 + 2}{2}})

ⓘ The numerator and the denominator are computed before the division:

1 + 2

3

\frac{d}{d x} ({(3 x - 12)}^{\frac{3}{2}})

We differentiate using:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

(where f =

3 x - 12

)

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} ({(3 x - 12)}^{\frac{3}{2}})

\frac{3}{2} \cdot \frac{d}{d x} (3 x - 12) \cdot \sqrt{3 x - 12}

\frac{3}{2} \cdot \frac{d}{d x} (3 x - 12) \cdot \sqrt{3 x - 12}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{3}{2} \cdot \frac{d}{d x} (3 x - 12) \cdot \sqrt{3 x - 12}

\frac{3 \cdot \frac{d}{d x} (3 x - 12) \cdot \sqrt{3 x - 12}}{2}

■The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (3 x - 12)

\frac{d}{d x} (3 x) + 0

\frac{3 \cdot (\frac{d}{d x} (3 x) + 0) \cdot \sqrt{3 x - 12}}{2}

ⓘ Adding 0 changes nothing:

\frac{d}{d x} (3 x) + 0

\frac{d}{d x} (3 x)

■A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d x} (3 x)

3

\frac{3 \cdot 3 \cdot \sqrt{3 x - 12}}{2}

ⅰ

3 \cdot 3

9

\frac{9 \cdot \sqrt{3 x - 12}}{2}

The computed derivative matches the original math expression up to a constant factor. Therefore, the integral is:

ⓘ The derivative of

\frac{2}{9} \cdot {(3 x - 12)}^{\frac{3}{2}}

\sqrt{3 x - 12}

, so

\frac{2}{9} \cdot {(3 x - 12)}^{\frac{3}{2}}

is an antiderivative of

\sqrt{3 x - 12}

\int \sqrt{3 x - 12} d x

\frac{2}{9} \cdot {(3 x - 12)}^{\frac{3}{2}}

\frac{2}{9} \cdot {(3 x - 12)}^{\frac{3}{2}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{2}{9} \cdot {(3 x - 12)}^{\frac{3}{2}}

\frac{2 \cdot {(3 x - 12)}^{\frac{3}{2}}}{9}

\frac{2 \cdot {(3 x - 12)}^{\frac{3}{2}}}{9}

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

\frac{2 \cdot {(3 x - 12)}^{\frac{3}{2}}}{9} + C