Calculus Examples

$\int 3.253 e^{0.1045 t} d t$

Step 1

Since $3.253$ is constant with respect to $t$ , move $3.253$ out of the integral.

$3.253 \int e^{0.1045 t} d t$

Step 2

Let $u = 0.1045 t$ . Then $d u = 0.1045 d t$ , so $\frac{1}{0.1045} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 0.1045 t$ . Find $\frac{d u}{d t}$ .

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Step 2.1.1

Differentiate $0.1045 t$ .

$\frac{d}{d t} [0.1045 t]$

Step 2.1.2

Since $0.1045$ is constant with respect to $t$ , the derivative of $0.1045 t$ with respect to $t$ is $0.1045 \frac{d}{d t} [t]$ .

$0.1045 \frac{d}{d t} [t]$

Step 2.1.3

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

$0.1045 \cdot 1$

Step 2.1.4

Multiply $0.1045$ by $1$ .

$0.1045$

Step 2.2

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

$3.253 \int e^{u} \frac{1}{0.1045} d u$

Step 3

Combine $e^{u}$ and $\frac{1}{0.1045}$ .

$3.253 \int \frac{e^{u}}{0.1045} d u$

Step 4

Since $\frac{1}{0.1045}$ is constant with respect to $u$ , move $\frac{1}{0.1045}$ out of the integral.

$3.253 (\frac{1}{0.1045} \int e^{u} d u)$

Step 5

Combine $\frac{1}{0.1045}$ and $3.253$ .

$\frac{3.253}{0.1045} \int e^{u} d u$

Step 6

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

$\frac{3.253}{0.1045} (e^{u} + C)$

Step 7

Rewrite $\frac{3.253}{0.1045} (e^{u} + C)$ as $31.1291866 e^{u} + C$ .

$31.1291866 e^{u} + C$

Step 8

Replace all occurrences of $u$ with $0.1045 t$ .

$31.1291866 e^{0.1045 t} + C$