Math / Calculus

QuestionDetermine the indefinite integral. $\begin{array}{c} \int 8 e^{7 x} d x \\ \int 8 e^{7 x} d x= \end{array}$ $\square$

Studdy Solution

STEP 1

What is this asking? We need to find the indefinite integral of $8e^{7x}$ with respect to $x$ .
Basically, we're looking for a function whose derivative is $8e^{7x}$ . Watch out! Remember the chain rule!
When differentiating $e$ to a power, the derivative of that power comes down as a factor.

STEP 2

1. Set up the integral
2. Find the integral

STEP 3

Alright, let's start with our integral: $\int 8e^{7x} dx$ .
We can rewrite this by pulling the constant 8 out front: $8 \int e^{7x} dx$ .
This makes things a little tidier!

STEP 4

Now, think about what function has a derivative of $e^{7x}$ .
If we just had $e^{7x}$ , its derivative would be $7e^{7x}$ because of the chain rule.
We'd bring the 7 down in front.

STEP 5

We want to end up with $e^{7x}$ after integrating, so let's try $\frac{1}{7}e^{7x}$ .
If we differentiate $\frac{1}{7}e^{7x}$ , we get $\frac{1}{7} \cdot 7e^{7x}$ , which simplifies to $1 \cdot e^{7x}$ or just $e^{7x}$ !
Perfect!

STEP 6

So, the integral of $e^{7x}$ is $\frac{1}{7}e^{7x}$ .
Don't forget the 8 we pulled out earlier!
We also need to add our constant of integration, which we'll call $+C$ .

STEP 7

Putting it all together, we have $8 \cdot \frac{1}{7}e^{7x} + C$ .
We can simplify this to $\frac{8}{7}e^{7x} + C$ .

STEP 8

Our final answer for the indefinite integral $\int 8e^{7x} dx$ is $\frac{8}{7}e^{7x} + C$ .

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QuestionDetermine the indefinite integral. $\begin{array}{c} \int 8 e^{7 x} d x \\ \int 8 e^{7 x} d x= \end{array}$ $\square$

Studdy Solution

STEP 1

What is this asking? We need to find the indefinite integral of $8e^{7x}$ with respect to $x$ .
Basically, we're looking for a function whose derivative is $8e^{7x}$ . Watch out! Remember the chain rule!
When differentiating $e$ to a power, the derivative of that power comes down as a factor.

STEP 2

1. Set up the integral
2. Find the integral

STEP 3

Alright, let's start with our integral: $\int 8e^{7x} dx$ .
We can rewrite this by pulling the constant 8 out front: $8 \int e^{7x} dx$ .
This makes things a little tidier!

STEP 4

Now, think about what function has a derivative of $e^{7x}$ .
If we just had $e^{7x}$ , its derivative would be $7e^{7x}$ because of the chain rule.
We'd bring the 7 down in front.

STEP 5

We want to end up with $e^{7x}$ after integrating, so let's try $\frac{1}{7}e^{7x}$ .
If we differentiate $\frac{1}{7}e^{7x}$ , we get $\frac{1}{7} \cdot 7e^{7x}$ , which simplifies to $1 \cdot e^{7x}$ or just $e^{7x}$ !
Perfect!

STEP 6

So, the integral of $e^{7x}$ is $\frac{1}{7}e^{7x}$ .
Don't forget the 8 we pulled out earlier!
We also need to add our constant of integration, which we'll call $+C$ .

STEP 7

Putting it all together, we have $8 \cdot \frac{1}{7}e^{7x} + C$ .
We can simplify this to $\frac{8}{7}e^{7x} + C$ .

STEP 8

Our final answer for the indefinite integral $\int 8e^{7x} dx$ is $\frac{8}{7}e^{7x} + C$ .

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QuestionDetermine the indefinite integral. ∫8e7xdx∫8e7xdx=\begin{array}{c} \int 8 e^{7 x} d x \\ \int 8 e^{7 x} d x= \end{array}∫8e7xdx∫8e7xdx=​ □\square□

Studdy Solution

STEP 1

STEP 2

1. Set up the integral2. Find the integral

STEP 3

Alright, let's **start** with our integral: ∫8e7xdx\int 8e^{7x} dx∫8e7xdx.We can **rewrite** this by pulling the constant **8** out front: 8∫e7xdx8 \int e^{7x} dx8∫e7xdx.This makes things a little tidier!

STEP 4

Now, think about what function has a derivative of e7xe^{7x}e7x.If we just had e7xe^{7x}e7x, its derivative would be 7e7x7e^{7x}7e7x because of the chain rule.We'd bring the **7** down in front.

STEP 5

We want to end up with e7xe^{7x}e7x after integrating, so let's try 17e7x\frac{1}{7}e^{7x}71​e7x.If we differentiate 17e7x\frac{1}{7}e^{7x}71​e7x, we get 17⋅7e7x\frac{1}{7} \cdot 7e^{7x}71​⋅7e7x, which simplifies to 1⋅e7x1 \cdot e^{7x}1⋅e7x or just e7xe^{7x}e7x!Perfect!

STEP 6

So, the integral of e7xe^{7x}e7x is 17e7x\frac{1}{7}e^{7x}71​e7x.Don't forget the **8** we pulled out earlier!We also need to add our constant of integration, which we'll call +C+C+C.

STEP 7

Putting it all together, we have 8⋅17e7x+C8 \cdot \frac{1}{7}e^{7x} + C8⋅71​e7x+C.We can simplify this to 87e7x+C\frac{8}{7}e^{7x} + C78​e7x+C.

STEP 8

Our **final answer** for the indefinite integral ∫8e7xdx\int 8e^{7x} dx∫8e7xdx is 87e7x+C\frac{8}{7}e^{7x} + C78​e7x+C.

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QuestionDetermine the indefinite integral. $\begin{array}{c} \int 8 e^{7 x} d x \\ \int 8 e^{7 x} d x= \end{array}$ $\square$

1. Set up the integral
2. Find the integral

Alright, let's start with our integral: $\int 8e^{7x} dx$ .
We can rewrite this by pulling the constant 8 out front: $8 \int e^{7x} dx$ .
This makes things a little tidier!

Now, think about what function has a derivative of $e^{7x}$ .
If we just had $e^{7x}$ , its derivative would be $7e^{7x}$ because of the chain rule.
We'd bring the 7 down in front.

We want to end up with $e^{7x}$ after integrating, so let's try $\frac{1}{7}e^{7x}$ .
If we differentiate $\frac{1}{7}e^{7x}$ , we get $\frac{1}{7} \cdot 7e^{7x}$ , which simplifies to $1 \cdot e^{7x}$ or just $e^{7x}$ !
Perfect!

So, the integral of $e^{7x}$ is $\frac{1}{7}e^{7x}$ .
Don't forget the 8 we pulled out earlier!
We also need to add our constant of integration, which we'll call $+C$ .

Putting it all together, we have $8 \cdot \frac{1}{7}e^{7x} + C$ .
We can simplify this to $\frac{8}{7}e^{7x} + C$ .

Our final answer for the indefinite integral $\int 8e^{7x} dx$ is $\frac{8}{7}e^{7x} + C$ .