Calculus Examples

$6^{2 t} + 2 t$

Step 1

Write $6^{2 t} + 2 t$ as a function.

$f (t) = 6^{2 t} + 2 t$

Step 2

The function $F (t)$ can be found by finding the indefinite integral of the derivative $f (t)$ .

$F (t) = \int f (t) d t$

Step 3

Set up the integral to solve.

$F (t) = \int 6^{2 t} + 2 t d t$

Step 4

Split the single integral into multiple integrals.

$\int 6^{2 t} d t + \int 2 t d t$

Step 5

Let $u = 2 t$ . Then $d u = 2 d t$ , so $\frac{1}{2} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 2 t$ . Find $\frac{d u}{d t}$ .

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Step 5.1.1

Differentiate $2 t$ .

$\frac{d}{d t} [2 t]$

Step 5.1.2

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

$2 \frac{d}{d t} [t]$

Step 5.1.3

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

$2 \cdot 1$

Step 5.1.4

Multiply $2$ by $1$ .

$2$

Step 5.2

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

$\int 6^{u} \cdot \frac{1}{2} d u + \int 2 t d t$

Step 6

Combine $6^{u}$ and $\frac{1}{2}$ .

$\int \frac{6^{u}}{2} d u + \int 2 t d t$

Step 7

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int 6^{u} d u + \int 2 t d t$

Step 8

The integral of $6^{u}$ with respect to $u$ is $\frac{6^{u}}{\ln (6)}$ .

$\frac{1}{2} (\frac{6^{u}}{\ln (6)} + C) + \int 2 t d t$

Step 9

Since $2$ is constant with respect to $t$ , move $2$ out of the integral.

$\frac{1}{2} (\frac{6^{u}}{\ln (6)} + C) + 2 \int t d t$

Step 10

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

$\frac{1}{2} (\frac{6^{u}}{\ln (6)} + C) + 2 (\frac{1}{2} t^{2} + C)$

Step 11

Simplify.

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Step 11.1

Combine $\frac{1}{2}$ and $t^{2}$ .

$\frac{1}{2} (\frac{6^{u}}{\ln (6)} + C) + 2 (\frac{t^{2}}{2} + C)$

Step 11.2

Simplify.

$\frac{1 \cdot 6^{u}}{2 \ln (6)} + 2 \frac{t^{2}}{2} + C$

Step 11.3

Simplify.

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Step 11.3.1

Multiply $6^{u}$ by $1$ .

$\frac{6^{u}}{2 \ln (6)} + 2 \frac{t^{2}}{2} + C$

Step 11.3.2

Combine $2$ and $\frac{t^{2}}{2}$ .

$\frac{6^{u}}{2 \ln (6)} + \frac{2 t^{2}}{2} + C$

Step 11.3.3

Cancel the common factor of $2$ .

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Step 11.3.3.1

Cancel the common factor.

$\frac{6^{u}}{2 \ln (6)} + \frac{2 t^{2}}{2} + C$

Step 11.3.3.2

Divide $t^{2}$ by $1$ .

$\frac{6^{u}}{2 \ln (6)} + t^{2} + C$

Step 12

Replace all occurrences of $u$ with $2 t$ .

$\frac{6^{2 t}}{2 \ln (6)} + t^{2} + C$

Step 13

Reorder terms.

$\frac{1}{2 \ln (6)} \cdot 6^{2 t} + t^{2} + C$

Step 14

The answer is the antiderivative of the function $f (t) = 6^{2 t} + 2 t$ .

$F (t) =$ $\frac{1}{2 \ln (6)} \cdot 6^{2 t} + t^{2} + C$