integral of v(u+v^2)^4 — step-by-step solution

The expression

v \cdot {(u + v^{2})}^{4}

contains several variables: the variable v is chosen for integration

1) Let's integrate:

\int v \cdot {(u + v^{2})}^{4} d v

For the math expression:

v \cdot {(u + v^{2})}^{4}

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

{(u + v^{2})}^{4 + 1}

2) Let's perform the differentiation:

\frac{d}{d v} ({(u + v^{2})}^{4 + 1})

ⓘ The exponent is computed before the power:

4 + 1

5

\frac{d}{d v} ({(u + v^{2})}^{5})

Apply the differentiation rule:

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

with f =

u + v^{2}

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d v} ({(u + v^{2})}^{5})

5 \cdot \frac{d}{d v} (u + v^{2}) \cdot {(u + v^{2})}^{4}

5 \cdot \frac{d}{d v} (u + v^{2}) \cdot {(u + v^{2})}^{4}

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

\frac{d}{d v} (u + v^{2})

\frac{d}{d v} (u) + \frac{d}{d v} (v^{2})

5 \cdot (\frac{d}{d v} (u) + \frac{d}{d v} (v^{2})) \cdot {(u + v^{2})}^{4}

ⓘ The derivative of a constant is zero:

\frac{d}{d v} (u)

0

■The derivative of x^n is n*x^(n-1):

\frac{d}{d v} (v^{2})

2 \cdot \frac{d}{d v} (v) \cdot v

5 \cdot (0 + 2 \cdot \frac{d}{d v} (v) \cdot v) \cdot {(u + v^{2})}^{4}

ⓘ Adding 0 changes nothing:

0 + 2 \cdot \frac{d}{d v} (v) \cdot v

2 \cdot \frac{d}{d v} (v) \cdot v

■The derivative of x is 1:

\frac{d}{d v} (v)

1

5 \cdot 2 \cdot 1 \cdot v \cdot {(u + v^{2})}^{4}

5 \cdot 2 v \cdot {(u + v^{2})}^{4}

ⅰ

5 \cdot 2

10

10 v \cdot {(u + v^{2})}^{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}{10} \cdot {(u + v^{2})}^{5}

v \cdot {(u + v^{2})}^{4}

, so

\frac{1}{10} \cdot {(u + v^{2})}^{5}

is an antiderivative of

v \cdot {(u + v^{2})}^{4}

\int v \cdot {(u + v^{2})}^{4} d v

\frac{1}{10} \cdot {(u + v^{2})}^{5}

\frac{1}{10} \cdot {(u + v^{2})}^{5}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{1}{10} \cdot {(u + v^{2})}^{5}

\frac{1 \cdot {(u + v^{2})}^{5}}{10}

\frac{{(u + v^{2})}^{5}}{10}

\frac{{(u + v^{2})}^{5}}{10}

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

\frac{{(u + v^{2})}^{5}}{10} + C