Calculus Examples

${(3 t + 4)}^{2.4}$

Step 1

Let $u = 3 t + 4$ . Then $d u = 3 d t$ , so $\frac{1}{3} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 3 t + 4$ . Find $\frac{d u}{d t}$ .

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Step 1.1.1

Differentiate $3 t + 4$ .

$\frac{d}{d t} [3 t + 4]$

Step 1.1.2

By the Sum Rule, the derivative of $3 t + 4$ with respect to $t$ is $\frac{d}{d t} [3 t] + \frac{d}{d t} [4]$ .

$\frac{d}{d t} [3 t] + \frac{d}{d t} [4]$

Step 1.1.3

Evaluate $\frac{d}{d t} [3 t]$ .

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Step 1.1.3.1

Since $3$ is constant with respect to $t$ , the derivative of $3 t$ with respect to $t$ is $3 \frac{d}{d t} [t]$ .

$3 \frac{d}{d t} [t] + \frac{d}{d t} [4]$

Step 1.1.3.2

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

$3 \cdot 1 + \frac{d}{d t} [4]$

Step 1.1.3.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d t} [4]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $4$ is constant with respect to $t$ , the derivative of $4$ with respect to $t$ is $0$ .

$3 + 0$

Step 1.1.4.2

Add $3$ and $0$ .

$3$

Step 1.2

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

$\int u^{2.4} \frac{1}{3} d u$

Step 2

Combine $u^{2.4}$ and $\frac{1}{3}$ .

$\int \frac{u^{2.4}}{3} d u$

Step 3

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int u^{2.4} d u$

Step 4

By the Power Rule, the integral of $u^{2.4}$ with respect to $u$ is $\frac{1}{3.4} u^{3.4}$ .

$\frac{1}{3} (\frac{1}{3.4} u^{3.4} + C)$

Step 5

Simplify.

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Step 5.1

Rewrite $\frac{1}{3} (\frac{1}{3.4} u^{3.4} + C)$ as $\frac{1}{3} (0.29411764 u^{3.4}) + C$ .

$\frac{1}{3} (0.29411764 u^{3.4}) + C$

Step 5.2

Simplify.

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Step 5.2.1

Combine $0.29411764$ and $\frac{1}{3}$ .

$\frac{0.29411764}{3} u^{3.4} + C$

Step 5.2.2

Combine $\frac{0.29411764}{3}$ and $u^{3.4}$ .

$\frac{0.29411764 u^{3.4}}{3} + C$

Step 6

Replace all occurrences of $u$ with $3 t + 4$ .

$\frac{0.29411764 {(3 t + 4)}^{3.4}}{3} + C$

Step 7

Reorder terms.

$\frac{0.29411764}{3} {(3 t + 4)}^{3.4} + C$