integrate 2x+3 — step-by-step solution

Let's integrate:

\int (2 x + 3) d x

ⅰ

\int (2 x + 3) d x

\int 2 x d x + \int 3 d x

\int 2 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 2 x d x

2 \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

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

ⓘ Multiplying a fraction by a number multiplies its numerator:

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

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

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

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

x^{2}

x^{2} + 3 x

You can write the expression in factored form:

x \cdot (x + 3)

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

x \cdot (x + 3) + C