integral of 6x+3 — step-by-step solution

Let's integrate:

\int (6 x + 3) d x

ⅰ

\int (6 x + 3) d x

\int 6 x d x + \int 3 d x

\int 6 x d x + \int 3 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 6 x d x

6 \cdot \frac{x^{2}}{2}

■The derivative of

3 x

3

, so

3 x

is an antiderivative of

3

\int 3 d x

3 x

6 \cdot \frac{x^{2}}{2} + 3 x

ⓘ Multiplying a fraction by a number multiplies its numerator:

6 \cdot \frac{x^{2}}{2}

\frac{6 x^{2}}{2}

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

\frac{6 x^{2}}{2}

3 x^{2}

3 x^{2} + 3 x

You can write the expression in factored form:

x \cdot (3 x + 3)

⇥Let's factor in

ℝ

(3 x + 3)

⇥We factor out

3

, the greatest common divisor of the coefficients:

3 x + 3

3 \cdot (x + 1)

⇥Factor identified:

x + 1

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

3 x \cdot (x + 1) + C