integrate the derivative of x^5 — step-by-step solution

⇥Let's perform the differentiation:

\frac{d}{d x} (x^{5})

⇥ⓘ The derivative of

x^{n}

n x^{n - 1}

\frac{d}{d x} (x^{5})

5 x^{4}

⇥

5 x^{4}

Let's integrate:

\int 5 x^{4} d x

ⓘ A constant factor stays in front of the integral. An antiderivative of

x^{n}

\frac{x^{n + 1}}{n + 1}

, for constant

n

with

n \neq -1

\int 5 x^{4} d x

5 \cdot \frac{x^{5}}{5}

5 \cdot \frac{x^{5}}{5}

ⓘ Multiplying a fraction by a number multiplies its numerator:

5 \cdot \frac{x^{5}}{5}

\frac{5 x^{5}}{5}

■To simplify a fraction, divide its numerator and denominator by the same nonzero number:

\frac{5 x^{5}}{5}

x^{5}

x^{5}

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

x^{5} + C