Calculus Examples

$\int_{0}^{1} (e^{t} + 5 e^{5 t}) d t$

Step 1

Remove parentheses.

$\int_{0}^{1} e^{t} + 5 e^{5 t} d t$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{1} e^{t} d t + \int_{0}^{1} 5 e^{5 t} d t$

Step 3

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

$e^{t}]_{0}^{1} + \int_{0}^{1} 5 e^{5 t} d t$

Step 4

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

$e^{t}]_{0}^{1} + 5 \int_{0}^{1} e^{5 t} d t$

Step 5

Let $u = 5 t$ . 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 5.1

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

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Step 5.1.1

Differentiate $5 t$ .

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

Step 5.1.2

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

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

Step 5.1.3

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

$5 \cdot 1$

Step 5.1.4

Multiply $5$ by $1$ .

$5$

Step 5.2

Substitute the lower limit in for $t$ in $u = 5 t$ .

$u_{lower} = 5 \cdot 0$

Step 5.3

Multiply $5$ by $0$ .

$u_{lower} = 0$

Step 5.4

Substitute the upper limit in for $t$ in $u = 5 t$ .

$u_{upper} = 5 \cdot 1$

Step 5.5

Multiply $5$ by $1$ .

$u_{upper} = 5$

Step 5.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 5$

Step 5.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$e^{t}]_{0}^{1} + 5 \int_{0}^{5} e^{u} \frac{1}{5} d u$

Step 6

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

$e^{t}]_{0}^{1} + 5 \int_{0}^{5} \frac{e^{u}}{5} d u$

Step 7

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

$e^{t}]_{0}^{1} + 5 (\frac{1}{5} \int_{0}^{5} e^{u} d u)$

Step 8

Simplify.

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Step 8.1

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

$e^{t}]_{0}^{1} + \frac{5}{5} \int_{0}^{5} e^{u} d u$

Step 8.2

Cancel the common factor of $5$ .

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Step 8.2.1

Cancel the common factor.

$e^{t}]_{0}^{1} + \frac{5}{5} \int_{0}^{5} e^{u} d u$

Step 8.2.2

Rewrite the expression.

$e^{t}]_{0}^{1} + 1 \int_{0}^{5} e^{u} d u$

Step 8.3

Multiply $\int_{0}^{5} e^{u} d u$ by $1$ .

$e^{t}]_{0}^{1} + \int_{0}^{5} e^{u} d u$

Step 9

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

$e^{t}]_{0}^{1} + e^{u}]_{0}^{5}$

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $e^{t}$ at $1$ and at $0$ .

$(e^{1}) - e^{0} + e^{u}]_{0}^{5}$

Step 10.2

Evaluate $e^{u}$ at $5$ and at $0$ .

$(e^{1}) - e^{0} + (e^{5}) - e^{0}$

Step 10.3

Simplify.

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Step 10.3.1

Simplify.

$e - e^{0} + (e^{5}) - e^{0}$

Step 10.3.2

Anything raised to $0$ is $1$ .

$e - 1 \cdot 1 + (e^{5}) - e^{0}$

Step 10.3.3

Multiply $- 1$ by $1$ .

$e - 1 + (e^{5}) - e^{0}$

Step 10.3.4

Anything raised to $0$ is $1$ .

$e - 1 + e^{5} - 1 \cdot 1$

Step 10.3.5

Multiply $- 1$ by $1$ .

$e - 1 + e^{5} - 1$

Step 10.3.6

Subtract $1$ from $- 1$ .

$e + e^{5} - 2$

Step 11

The result can be shown in multiple forms.

Exact Form:

$e + e^{5} - 2$

Decimal Form:

$149.13144093 \dots$

Step 12