integral of 12x^2ln(x) — step-by-step solution

1) Let's compute the integral:

\int 12 x^{2} \cdot \ln (x) d x

⇥Take the constant factor

12

outside the integral. For

12 x^{2} \cdot \ln (x)

, we use integration by parts:

\int 12 \cdot u1 \cdot v2' d x = 12 \cdot (u1 \cdot v2 - \int u1' \cdot v2 d x)

, with u1 =

\ln (x)

and v2' =

x^{2}

⇥1.1) Let's perform the differentiation u1:

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

⇥ⓘ The derivative of ln(x) is 1/x:

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

\frac{1}{x}

⇥

\frac{1}{x}

⇥1.2) Let's integrate v2':

\int x^{2} d x

⇥ⓘ An antiderivative of x^n is x^(n+1)/(n+1), for n ≠ -1:

\int x^{2} d x

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

⇥

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

⇥1.3) Let's evaluate

u1' \cdot v2

\frac{1}{x} \cdot \frac{x^{3}}{3}

⇥ⓘ To multiply fractions, multiply the numerators together and the denominators together:

\frac{1}{x} \cdot \frac{x^{3}}{3}

\frac{1 \cdot x^{3}}{x \cdot 3}

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

■The original denominator must remain nonzero. The fraction is simplified by dividing the numerator and the denominator by their common factor:

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

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

⇥

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

⇥1.4) Let's calculate

12 \cdot (u1 \cdot v2 - \int u1' \cdot v2 d x)

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

⇥ⓘ Multiplying a fraction by a term multiplies its numerator:

\ln (x) \cdot \frac{x^{3}}{3}

\frac{\ln (x) \cdot x^{3}}{3}

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

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

- \frac{1}{3} \cdot \frac{x^{3}}{3}

⇥

12 \cdot (\frac{\ln (x) \cdot x^{3}}{3} - \frac{1}{3} \cdot \frac{x^{3}}{3})

⇥ⓘ To multiply fractions, multiply the numerators together and the denominators together:

\frac{1}{3} \cdot \frac{x^{3}}{3}

\frac{1 \cdot x^{3}}{3 \cdot 3}

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

⇥

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

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

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

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

⇥

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

⇥ⓘ Multiplying a fraction by a number multiplies its numerator:

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

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

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

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

\frac{4 \cdot (3 \cdot \ln (x) \cdot x^{3} - x^{3})}{3}

⇥

\frac{4 \cdot (3 \cdot \ln (x) \cdot x^{3} - x^{3})}{3}

⇥ⓘ What the terms have in common is factored out, so that the fraction can be simplified:

\frac{4 \cdot (3 \cdot \ln (x) \cdot x^{3} - x^{3})}{3}

\frac{4 x^{3} \cdot (3 \cdot \ln (x) - 1)}{3}

⇥

\frac{4 x^{3} \cdot (3 \cdot \ln (x) - 1)}{3}

Finally you can finish the integration:

ⅰ

\int 12 x^{2} \cdot \ln (x) d x

\frac{4 x^{3} \cdot (3 \cdot \ln (x) - 1)}{3}

\frac{4 x^{3} \cdot (3 \cdot \ln (x) - 1)}{3}

ⓘ Expand by multiplying each term inside the parentheses:

4 x^{3} \cdot (3 \cdot \ln (x) - 1)

12 x^{3} \cdot \ln (x) + 4 \cdot (- x^{3})

\frac{12 x^{3} \cdot \ln (x) + 4 \cdot (- x^{3})}{3}

ⓘ The minus sign is placed in front of the product:

4 \cdot (- x^{3})

- 4 x^{3}

\frac{12 x^{3} \cdot \ln (x) - 4 x^{3}}{3}

ⓘ A fraction whose numerator is a sum splits term by term, (a + b)/c = a/c + b/c:

\frac{12 x^{3} \cdot \ln (x) - 4 x^{3}}{3}

4 \cdot x^{3} \cdot \ln (x) - \frac{4 x^{3}}{3}

4 \cdot x^{3} \cdot \ln (x) - \frac{4 x^{3}}{3}

We factor out

4

, the greatest common divisor of the coefficients:

4 x^{3} \cdot \ln (x) - \frac{4 x^{3}}{3}

4 \cdot (x^{3} \cdot \ln (x) - \frac{x^{3}}{3})

Factor identified:

x^{3} \cdot \ln (x) - \frac{x^{3}}{3}

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

4 \cdot (x^{3} \cdot \ln (x) - \frac{x^{3}}{3}) + C