Calculus Examples

$\int 5 e^{5 t + 2} d t$

Step 1

Since $5$ is constant with respect to $t$ , move $5$ out of the integral.

$5 \int e^{5 t + 2} d t$

Step 2

Let $u = 5 t + 2$ . Then $d u = 5 d t$ , so $\frac{1}{5} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 5 t + 2$ . Find $\frac{d u}{d t}$ .

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Step 2.1.1

Differentiate $5 t + 2$ .

$\frac{d}{d t} [5 t + 2]$

Step 2.1.2

By the Sum Rule, the derivative of $5 t + 2$ with respect to $t$ is $\frac{d}{d t} [5 t] + \frac{d}{d t} [2]$ .

$\frac{d}{d t} [5 t] + \frac{d}{d t} [2]$

Step 2.1.3

Evaluate $\frac{d}{d t} [5 t]$ .

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Step 2.1.3.1

Since $5$ is constant with respect to $t$ , the derivative of $5 t$ with respect to $t$ is $5 \frac{d}{d t} [t]$ .

$5 \frac{d}{d t} [t] + \frac{d}{d t} [2]$

Step 2.1.3.2

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

$5 \cdot 1 + \frac{d}{d t} [2]$

Step 2.1.3.3

Multiply $5$ by $1$ .

$5 + \frac{d}{d t} [2]$

Step 2.1.4

Differentiate using the Constant Rule.

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Step 2.1.4.1

Since $2$ is constant with respect to $t$ , the derivative of $2$ with respect to $t$ is $0$ .

$5 + 0$

Step 2.1.4.2

Add $5$ and $0$ .

$5$

Step 2.2

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

$5 \int e^{u} \frac{1}{5} d u$

Step 3

Combine $e^{u}$ and $\frac{1}{5}$ .

$5 \int \frac{e^{u}}{5} d u$

Step 4

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

$5 (\frac{1}{5} \int e^{u} d u)$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{5}$ and $5$ .

$\frac{5}{5} \int e^{u} d u$

Step 5.2

Cancel the common factor of $5$ .

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Step 5.2.1

Cancel the common factor.

$\frac{5}{5} \int e^{u} d u$

Step 5.2.2

Rewrite the expression.

$1 \int e^{u} d u$

Step 5.3

Multiply $\int e^{u} d u$ by $1$ .

$\int e^{u} d u$

Step 6

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$e^{u} + C$

Step 7

Replace all occurrences of $u$ with $5 t + 2$ .

$e^{5 t + 2} + C$