Question

What is ∫x(x+3)12dx?

Answer 1

Answer : $4 x{}^{3} + 18x {}^{2} + C$

Step by step explanation :

$\int \times (x + 3) \times 12 \: dx$

Use property of integral

$\int a \times f(x)dx = a \times \int f(x) \: dx$

$= > 12 \times \int x \times (x + 3) \: dx$

Distribute x through the parenthesis by multiplying each term in the parenthesis by x

$= > 12 \times \int x \times x + 3x \: dx$

Calculate the product

$= > 12 \times \int x {}^{2} + 3x \: dx$

Use property of integral

$\int f(x) + - g(x) \: dx = \int f(x)dx + - \int g(x) \: dx$

$= > 12( \int x {}^{2} dx + \int 3x \: dx$

Using $\int x {}^{n} dx = \frac{x {}^{n + 1} }{n + 1} \: \:, \: n≠ - 1$ solve the integral.

$= > 12( \frac{x {}^{2 + 1} }{2 + 1} +\int 3x \: dx)$

Add the numbers

$= > 12( \frac{x {}^{3} }{3} + \int 3x \: dx)$

Calculate the indefinite integral

Use property of integral

$\int \: a \times f(x) \: dx = a \times \int f(x) \: dx$

$= > 12( \frac{x {}^{3} }{3} + 3 \times \int x \: dx)$

Using $c\intx \: dx = \frac{x {}^{2} }{2}$ , solve the integral.

$= > 12( \frac{x {}^{3} }{3} + 3 \times \frac{x {}^{2} }{2} )$

Calculate the product.

$= > 12( \frac{x {}^{3} }{3} + \frac{3x {}^{2} }{2} )$

Distribute 12 through the parenthesis by multiplying each term in the parenthesis by 12

$= > 12 \times \frac{x {}^{3} }{3} + 12 \times \frac{3x {}^{2} }{2}$

Reduce the numbers with the greatest common divisor 3

$= > 4x {}^{3} + 12 \times \frac{3x {}^{2} }{2}$

Reduce numbers with the greatest common divisor 2

$= > 4x {}^{3} + 6 \times 3x {}^{2}$

Calculate the product

$= > 4x {}^{3} + 18x {}^{2}$

Add the constant of integration.

$= > 4x {}^{3} + 18x {}^{2} + C$

Answer 2

Answer:

4x³ + 18x² + C

Step-by-step explanation:

⇒ ∫ 12x ( x + 3 ) dx

Multiplying 12x inside the bracket we get,

⇒ ∫ ( 12x² + 36x ) dx

Splitting the integrals we get,

⇒ ∫ 12x². dx + ∫ 36x . dx

Integrating we get,

⇒ 12x³/3 + 36x²/2 + C

⇒ 4x³ + 18x² + C

This is the required solution.

Hope it helped !!