Calculus Examples

$\int_{0}^{10} 3 e^{0.7 (10 - t)} d t$

Step 1

Since $3$ is constant with respect to $t$ , move $3$ out of the integral.

$3 \int_{0}^{10} e^{0.7 (10 - t)} d t$

Step 2

Let $u = 0.7 (10 - t)$ . Then $d u = - 0.7 d t$ , so $\frac{1}{- 0.7} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 0.7 (10 - t)$ . Find $\frac{d u}{d t}$ .

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Step 2.1.1

Differentiate $0.7 (10 - t)$ .

$\frac{d}{d t} [0.7 (10 - t)]$

Step 2.1.2

Since $0.7$ is constant with respect to $t$ , the derivative of $0.7 (10 - t)$ with respect to $t$ is $0.7 \frac{d}{d t} [10 - t]$ .

$0.7 \frac{d}{d t} [10 - t]$

Step 2.1.3

By the Sum Rule, the derivative of $10 - t$ with respect to $t$ is $\frac{d}{d t} [10] + \frac{d}{d t} [- t]$ .

$0.7 (\frac{d}{d t} [10] + \frac{d}{d t} [- t])$

Step 2.1.4

Since $10$ is constant with respect to $t$ , the derivative of $10$ with respect to $t$ is $0$ .

$0.7 (0 + \frac{d}{d t} [- t])$

Step 2.1.5

Add $0$ and $\frac{d}{d t} [- t]$ .

$0.7 \frac{d}{d t} [- t]$

Step 2.1.6

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

$0.7 (- \frac{d}{d t} [t])$

Step 2.1.7

Multiply $- 1$ by $0.7$ .

$- 0.7 \frac{d}{d t} [t]$

Step 2.1.8

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

$- 0.7 \cdot 1$

Step 2.1.9

Multiply $- 0.7$ by $1$ .

$- 0.7$

Step 2.2

Substitute the lower limit in for $t$ in $u = 0.7 (10 - t)$ .

$u_{lower} = 0.7 (10 - 0)$

Step 2.3

Simplify.

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Step 2.3.1

Subtract $0$ from $10$ .

$u_{lower} = 0.7 \cdot 10$

Step 2.3.2

Multiply $0.7$ by $10$ .

$u_{lower} = 7$

Step 2.4

Substitute the upper limit in for $t$ in $u = 0.7 (10 - t)$ .

$u_{upper} = 0.7 (10 - 1 \cdot 10)$

Step 2.5

Simplify.

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Step 2.5.1

Multiply $- 1$ by $10$ .

$u_{upper} = 0.7 (10 - 10)$

Step 2.5.2

Subtract $10$ from $10$ .

$u_{upper} = 0.7 \cdot 0$

Step 2.5.3

Multiply $0.7$ by $0$ .

$u_{upper} = 0$

Step 2.6

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

$u_{lower} = 7$

$u_{upper} = 0$

Step 2.7

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

$3 \int_{7}^{0} e^{u} \frac{1}{- 0.7} d u$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$3 \int_{7}^{0} e^{u} (- \frac{1}{0.7}) d u$

Step 3.2

Combine $e^{u}$ and $\frac{1}{0.7}$ .

$3 \int_{7}^{0} - \frac{e^{u}}{0.7} d u$

Step 4

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$3 (- \int_{7}^{0} \frac{e^{u}}{0.7} d u)$

Step 5

Multiply $- 1$ by $3$ .

$- 3 \int_{7}^{0} \frac{e^{u}}{0.7} d u$

Step 6

Since $\frac{1}{0.7}$ is constant with respect to $u$ , move $\frac{1}{0.7}$ out of the integral.

$- 3 (\frac{1}{0.7} \int_{7}^{0} e^{u} d u)$

Step 7

Simplify.

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Step 7.1

Combine $\frac{1}{0.7}$ and $- 3$ .

$\frac{- 3}{0.7} \int_{7}^{0} e^{u} d u$

Step 7.2

Move the negative in front of the fraction.

$- \frac{3}{0.7} \int_{7}^{0} e^{u} d u$

Step 8

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

$- \frac{3}{0.7} e^{u}]_{7}^{0}$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $e^{u}$ at $0$ and at $7$ .

$- \frac{3}{0.7} ((e^{0}) - e^{7})$

Step 9.2

Anything raised to $0$ is $1$ .

$- \frac{3}{0.7} (1 - e^{7})$

Step 10

Simplify.

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Step 10.1

Divide $3$ by $0.7$ .

$- 1 \cdot 4.\overline{285714} (1 - e^{7})$

Step 10.2

Multiply $- 1$ by $4.\overline{285714}$ .

$- 4.\overline{285714} (1 - e^{7})$

Step 10.3

Multiply $- 4.\overline{285714}$ by $1 - e^{7}$ .

$4695.57067897$

Step 11