Calculus Examples

$s' (t) = t {(t - 1)}^{2}$

Step 1

The function $s (t)$ can be found by evaluating the indefinite integral of the derivative $s' (t)$ .

$s (t) = \int s' (t) d t$

Step 2

Let $u = t - 1$ . Then $d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = t - 1$ . Find $\frac{d u}{d t}$ .

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Step 2.1.1

Differentiate $t - 1$ .

$\frac{d}{d t} [t - 1]$

Step 2.1.2

By the Sum Rule, the derivative of $t - 1$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [- 1]$ .

$\frac{d}{d t} [t] + \frac{d}{d t} [- 1]$

Step 2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d t} [- 1]$

Step 2.1.4

Since $- 1$ is constant with respect to $t$ , the derivative of $- 1$ with respect to $t$ is $0$ .

$1 + 0$

Step 2.1.5

Add $1$ and $0$ .

$1$

Step 2.2

Rewrite the problem using $u$ and $d u$ .

$\int (u + 1) u^{2} d u$

Step 3

Expand $(u + 1) u^{2}$ .

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Step 3.1

Apply the distributive property.

$\int u \cdot u^{2} + 1 u^{2} d u$

Step 3.2

Raise $u$ to the power of $1$ .

$\int u^{1} u^{2} + 1 u^{2} d u$

Step 3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int u^{1 + 2} + 1 u^{2} d u$

Step 3.4

Add $1$ and $2$ .

$\int u^{3} + 1 u^{2} d u$

Step 3.5

Multiply $u^{2}$ by $1$ .

$\int u^{3} + u^{2} d u$

Step 4

Split the single integral into multiple integrals.

$\int u^{3} d u + \int u^{2} d u$

Step 5

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

$\frac{1}{4} u^{4} + C + \int u^{2} d u$

Step 6

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$\frac{1}{4} u^{4} + C + \frac{1}{3} u^{3} + C$

Step 7

Simplify.

$\frac{1}{4} u^{4} + \frac{1}{3} u^{3} + C$

Step 8

Replace all occurrences of $u$ with $t - 1$ .

$\frac{1}{4} {(t - 1)}^{4} + \frac{1}{3} {(t - 1)}^{3} + C$

Step 9

The function $s$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

$s (t) = \frac{1}{4} \cdot {(t - 1)}^{4} + \frac{1}{3} \cdot {(t - 1)}^{3} + C$