Math / Calculus

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

Studdy Solution

STEP 1

What is this asking? We need to find the definite integral of $e^{4t}$ from $a$ to $b$ . Watch out! Don't forget about the chain rule when integrating!
Also, remember to carefully evaluate the integral at the bounds of integration.

STEP 2

1. Find the indefinite integral.
2. Evaluate the definite integral.

STEP 3

Let's start by finding the indefinite integral of $e^{4t}$ with respect to $t$ .
We write this as: $\int e^{4t} \, dt$

STEP 4

To handle the $4t$ in the exponent, we'll use u-substitution.
Let $u = 4t$ .
Then, the derivative of $u$ with respect to $t$ is $\frac{du}{dt} = 4$ .
So, $dt = \frac{1}{4} du$ .
Now we can rewrite our integral: $\int e^{u} \cdot \frac{1}{4} \, du$

STEP 5

We can pull the constant $\frac{1}{4}$ out front: $\frac{1}{4} \int e^{u} \, du$ Now, the integral of $e^u$ with respect to $u$ is simply $e^u$ .
So we have: $\frac{1}{4} e^{u} + C$ Don't forget the constant of integration, $C$ , since we're dealing with an indefinite integral!

STEP 6

Now, let's substitute back $u = 4t$ to get our indefinite integral in terms of $t$ : $\frac{1}{4} e^{4t} + C$

STEP 7

To evaluate the definite integral from $a$ to $b$ , we'll use the Fundamental Theorem of Calculus.
This tells us to subtract the value of the integral at the lower bound, $a$ , from the value at the upper bound, $b$ .

STEP 8

First, let's plug in $t = b$ : $\frac{1}{4} e^{4b}$

STEP 9

Next, let's plug in $t = a$ : $\frac{1}{4} e^{4a}$

STEP 10

Finally, we subtract the result from step 2.2.3. from the result from step 2.2.2.: $\frac{1}{4} e^{4b} - \frac{1}{4} e^{4a}$

STEP 11

Our final answer for the definite integral is: $\frac{1}{4} e^{4b} - \frac{1}{4} e^{4a}$

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