Question

Use a symbolic integration utility to find the indefinite integral. $$ \int(x+1)(3 x-2) d x $$

Short answer

The indefinite integral of the function \( (x+1)(3x-2) \) is \( x^3 + \frac{x^2}{2} - 2x + C \)

Step-by-step solution

Expand the integrand

Start by expanding the product within the integrand to simplify the integral. It results in: \[ \int x(3x-2) + 1(3x-2) dx = \int (3x^2 - 2x + 3x - 2) dx \] which further simplifies to: \[ \int (3x^2 + x - 2) dx \]

Separate integrals according to terms

Separate the integral to different terms which makes it easier to integrate: \[ \int 3x^2 dx + \int x dx - \int 2 dx \]

Integrate each term separately

Next, integrate each of these terms separately: \[ \frac{3}{3} \int x^2 dx + \frac{1}{2} \int 2x dx - 2 \int dx = x^3 + \frac{x^2}{2} - 2x + C \], where C is the constant of integration.