integrate x twice — step-by-step solution

1) Let's integrate:

\int x d x

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

\int x d x

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

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

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

\frac{x^{2}}{2} + C

2) Let's compute the integral:

\int (\frac{x^{2}}{2} + C) d x

ⅰ

\int (\frac{x^{2}}{2} + C) d x

\int \frac{x^{2}}{2} d x + \int C d x

\int \frac{x^{2}}{2} d x + \int C 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 \frac{x^{2}}{2} d x

\frac{1}{2} \cdot \frac{x^{3}}{3}

■The derivative of

C x

C

, so

C x

is an antiderivative of

C

\int C d x

C x

\frac{1}{2} \cdot \frac{x^{3}}{3} + C x

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

\frac{1}{2} \cdot \frac{x^{3}}{3}

\frac{1 \cdot x^{3}}{2 \cdot 3}

\frac{x^{3}}{6}

\frac{x^{3}}{6} + C x

▶Indefinite integrals are defined up to an additive constant C1, so this yields:

\frac{x^{3}}{6} + C x + C1

, with C and C1 being 2 constants