Calculus Examples

$\int 45 e^{0.27465 t} d t$

Step 1

Since $45$ is constant with respect to $t$ , move $45$ out of the integral.

$45 \int e^{0.27465 t} d t$

Step 2

Let $u = 0.27465 t$ . Then $d u = 0.27465 d t$ , so $\frac{1}{0.27465} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 0.27465 t$ . Find $\frac{d u}{d t}$ .

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Step 2.1.1

Differentiate $0.27465 t$ .

$\frac{d}{d t} [0.27465 t]$

Step 2.1.2

Since $0.27465$ is constant with respect to $t$ , the derivative of $0.27465 t$ with respect to $t$ is $0.27465 \frac{d}{d t} [t]$ .

$0.27465 \frac{d}{d t} [t]$

Step 2.1.3

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

$0.27465 \cdot 1$

Step 2.1.4

Multiply $0.27465$ by $1$ .

$0.27465$

Step 2.2

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

$45 \int e^{u} \frac{1}{0.27465} d u$

Step 3

Combine $e^{u}$ and $\frac{1}{0.27465}$ .

$45 \int \frac{e^{u}}{0.27465} d u$

Step 4

Since $\frac{1}{0.27465}$ is constant with respect to $u$ , move $\frac{1}{0.27465}$ out of the integral.

$45 (\frac{1}{0.27465} \int e^{u} d u)$

Step 5

Combine $\frac{1}{0.27465}$ and $45$ .

$\frac{45}{0.27465} \int e^{u} d u$

Step 6

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

$\frac{45}{0.27465} (e^{u} + C)$

Step 7

Rewrite $\frac{45}{0.27465} (e^{u} + C)$ as $163.8448935 e^{u} + C$ .

$163.8448935 e^{u} + C$

Step 8

Replace all occurrences of $u$ with $0.27465 t$ .

$163.8448935 e^{0.27465 t} + C$