Calculus Examples

$\int_{a}^{b} 5 e^{4 t} d t$

Step 1

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

$5 \int_{a}^{b} e^{4 t} d t$

Step 2

Let $u = 4 t$ . Then $d u = 4 d t$ , so $\frac{1}{4} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 4 t$ . Find $\frac{d u}{d t}$ .

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Step 2.1.1

Differentiate $4 t$ .

$\frac{d}{d t} [4 t]$

Step 2.1.2

Since $4$ is constant with respect to $t$ , the derivative of $4 t$ with respect to $t$ is $4 \frac{d}{d t} [t]$ .

$4 \frac{d}{d t} [t]$

Step 2.1.3

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

$4 \cdot 1$

Step 2.1.4

Multiply $4$ by $1$ .

$4$

Step 2.2

Substitute the lower limit in for $t$ in $u = 4 t$ .

$u_{lower} = 4 a$

Step 2.3

Substitute the upper limit in for $t$ in $u = 4 t$ .

$u_{upper} = 4 b$

Step 2.4

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

$u_{lower} = 4 a$

$u_{upper} = 4 b$

Step 2.5

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

$5 \int_{4 a}^{4 b} e^{u} \frac{1}{4} d u$

Step 3

Combine $e^{u}$ and $\frac{1}{4}$ .

$5 \int_{4 a}^{4 b} \frac{e^{u}}{4} d u$

Step 4

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$5 (\frac{1}{4} \int_{4 a}^{4 b} e^{u} d u)$

Step 5

Combine $\frac{1}{4}$ and $5$ .

$\frac{5}{4} \int_{4 a}^{4 b} e^{u} d u$

Step 6

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

$\frac{5}{4} e^{u}]_{4 a}^{4 b}$

Step 7

Evaluate $e^{u}$ at $4 b$ and at $4 a$ .

$\frac{5}{4} (e^{4 b} - e^{4 a})$