Calculus Examples

$\frac{e^{\frac{5}{t}}}{3 t^{2}}$

Step 1

Write $\frac{e^{\frac{5}{t}}}{3 t^{2}}$ as a function.

$f (t) = \frac{e^{\frac{5}{t}}}{3 t^{2}}$

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.

$F (t) = \int \frac{e^{\frac{5}{t}}}{3 t^{2}} d t$

Step 4

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

$\frac{1}{3} \int \frac{e^{\frac{5}{t}}}{t^{2}} d t$

Step 5

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

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Step 5.1

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

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Step 5.1.1

Differentiate $\frac{5}{t}$ .

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

Step 5.1.2

Since $5$ is constant with respect to $t$ , the derivative of $\frac{5}{t}$ with respect to $t$ is $5 \frac{d}{d t} [\frac{1}{t}]$ .

$5 \frac{d}{d t} [\frac{1}{t}]$

Step 5.1.3

Rewrite $\frac{1}{t}$ as $t^{- 1}$ .

$5 \frac{d}{d t} [t^{- 1}]$

Step 5.1.4

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

$5 (- t^{- 2})$

Step 5.1.5

Multiply $- 1$ by $5$ .

$- 5 t^{- 2}$

Step 5.1.6

Simplify.

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Step 5.1.6.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 5 \frac{1}{t^{2}}$

Step 5.1.6.2

Combine terms.

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Step 5.1.6.2.1

Combine $- 5$ and $\frac{1}{t^{2}}$ .

$\frac{- 5}{t^{2}}$

Step 5.1.6.2.2

Move the negative in front of the fraction.

$- \frac{5}{t^{2}}$

Step 5.2

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

$\frac{1}{3} \int - \frac{1}{5} e^{u} d u$

Step 6

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

$\frac{1}{3} (- \frac{1}{5} \int e^{u} d u)$

Step 7

Simplify.

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Step 7.1

Multiply $\frac{1}{3}$ by $\frac{1}{5}$ .

$- \frac{1}{3 \cdot 5} \int e^{u} d u$

Step 7.2

Multiply $3$ by $5$ .

$- \frac{1}{15} \int e^{u} d u$

Step 8

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

$- \frac{1}{15} (e^{u} + C)$

Step 9

Simplify.

$- \frac{1}{15} e^{u} + C$

Step 10

Replace all occurrences of $u$ with $\frac{5}{t}$ .

$- \frac{1}{15} e^{\frac{5}{t}} + C$

Step 11

The answer is the antiderivative of the function $f (t) = \frac{e^{\frac{5}{t}}}{3 t^{2}}$ .

$F (t) =$ $- \frac{1}{15} e^{\frac{5}{t}} + C$