twice the integral of x — step-by-step solution

⇥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

Let's evaluate:

2 \cdot (\frac{x^{2}}{2} + C)

Let's calculate:

2 \cdot (\frac{x^{2}}{2} + C)

ⓘ Expand by multiplying each term inside the parentheses:

2 \cdot (\frac{x^{2}}{2} + C)

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

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

ⓘ 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} + 2 C

ⓘ

2 C

only depends on constants: it is still an arbitrary constant, which is written

C

▶Result:

x^{2} + C

, where C is a constant