integral of (z+3)^4 — step-by-step solution

1) Let's integrate:

\int {(z + 3)}^{4} d z

For the math expression:

{(z + 3)}^{4}

, since (a^n)' = n*a'*a^(n - 1), we try differentiating

{(z + 3)}^{4 + 1}

2) Let's perform the differentiation:

\frac{d}{d z} ({(z + 3)}^{4 + 1})

ⓘ The exponent is computed before the power:

4 + 1

5

\frac{d}{d z} ({(z + 3)}^{5})

We differentiate using:

\frac{d}{d z} (f^{n}) = n \cdot \frac{d}{d z} (f) \cdot f^{n - 1}

(where f =

z + 3

)

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d z} ({(z + 3)}^{5})

5 \cdot \frac{d}{d z} (z + 3) \cdot {(z + 3)}^{4}

5 \cdot \frac{d}{d z} (z + 3) \cdot {(z + 3)}^{4}

ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d z} (z + 3)

1 + 0

5 \cdot (1 + 0) \cdot {(z + 3)}^{4}

ⓘ Adding 0 changes nothing:

1 + 0

1

5 \cdot {(z + 3)}^{4}

You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

ⓘ The derivative of

\frac{1}{5} \cdot {(z + 3)}^{5}

{(z + 3)}^{4}

, so

\frac{1}{5} \cdot {(z + 3)}^{5}

is an antiderivative of

{(z + 3)}^{4}

\int {(z + 3)}^{4} d z

\frac{1}{5} \cdot {(z + 3)}^{5}

\frac{1}{5} \cdot {(z + 3)}^{5}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{1}{5} \cdot {(z + 3)}^{5}

\frac{1 \cdot {(z + 3)}^{5}}{5}

\frac{{(z + 3)}^{5}}{5}

\frac{{(z + 3)}^{5}}{5}

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

\frac{{(z + 3)}^{5}}{5} + C