Math / Calculus

QuestionFind the most general antiderivative by evaluating the following indefinite integral: $\int \frac{6}{x} d x=$ $\square$
NOTE: The general antiderivative should contain an arbitrary constant.

Studdy Solution

STEP 1

What is this asking? We need to find the most general antiderivative of $\frac{6}{x}$ .
Basically, we're looking for a function whose derivative is $\frac{6}{x}$ , and we need to make sure we include the "+ C"! Watch out! Don't forget about the special case when dealing with $\frac{1}{x}$ !
Its antiderivative isn't a power rule situation.

STEP 2

1. Rewrite the integral
2. Apply the constant multiple rule
3. Integrate

STEP 3

Alright, let's rewrite our integral to make it super clear what we're working with.
We can pull that constant 6 right out front!
This makes things easier to see.
$\int \frac{6}{x} dx = \int 6 \cdot \frac{1}{x} dx$

STEP 4

Why can we do this?
Because it's a constant multiple!
Remember, constants just scale things up or down, they don't change the fundamental nature of the function.

STEP 5

Now, we can apply the constant multiple rule.
This rule says that the integral of a constant times a function is the same as the constant times the integral of the function.
So, we get:
$\int 6 \cdot \frac{1}{x} dx = 6 \cdot \int \frac{1}{x} dx$

STEP 6

Think of it like this: if you're finding the area under a curve that's 6 times taller than another curve, the total area will be 6 times bigger!

STEP 7

Now for the main event: integrating!
The integral of $\frac{1}{x}$ is the natural logarithm of the absolute value of $x$ , written as $\ln|x|$ .
Why the absolute value?
Because we can't take the logarithm of a negative number!
So, we have:
$6 \cdot \int \frac{1}{x} dx = 6 \cdot \ln|x|$

STEP 8

Almost there!
The most general antiderivative needs a constant of integration, often denoted as "+ C".
This constant represents the fact that there are infinitely many functions with the same derivative, all differing by a constant value.
So, our final answer is:
$6 \cdot \ln|x| + C$

STEP 9

$6 \ln|x| + C$

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QuestionFind the most general antiderivative by evaluating the following indefinite integral: $\int \frac{6}{x} d x=$ $\square$
NOTE: The general antiderivative should contain an arbitrary constant.

Studdy Solution

STEP 1

STEP 2

1. Rewrite the integral
2. Apply the constant multiple rule
3. Integrate

STEP 3

Alright, let's rewrite our integral to make it super clear what we're working with.
We can pull that constant 6 right out front!
This makes things easier to see.
$\int \frac{6}{x} dx = \int 6 \cdot \frac{1}{x} dx$

STEP 4

Why can we do this?
Because it's a constant multiple!
Remember, constants just scale things up or down, they don't change the fundamental nature of the function.

STEP 5

Now, we can apply the constant multiple rule.
This rule says that the integral of a constant times a function is the same as the constant times the integral of the function.
So, we get:
$\int 6 \cdot \frac{1}{x} dx = 6 \cdot \int \frac{1}{x} dx$

STEP 6

Think of it like this: if you're finding the area under a curve that's 6 times taller than another curve, the total area will be 6 times bigger!

STEP 7

Now for the main event: integrating!
The integral of $\frac{1}{x}$ is the natural logarithm of the absolute value of $x$ , written as $\ln|x|$ .
Why the absolute value?
Because we can't take the logarithm of a negative number!
So, we have:
$6 \cdot \int \frac{1}{x} dx = 6 \cdot \ln|x|$

STEP 8

Almost there!
The most general antiderivative needs a constant of integration, often denoted as "+ C".
This constant represents the fact that there are infinitely many functions with the same derivative, all differing by a constant value.
So, our final answer is:
$6 \cdot \ln|x| + C$

STEP 9

$6 \ln|x| + C$

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QuestionFind the most general antiderivative by evaluating the following indefinite integral: ∫6xdx=\int \frac{6}{x} d x=∫x6​dx= □\square□NOTE: The general antiderivative should contain an arbitrary constant.

Studdy Solution

STEP 1

STEP 2

1. Rewrite the integral2. Apply the constant multiple rule3. Integrate

STEP 3

Alright, let's **rewrite** our integral to make it super clear what we're working with.We can pull that constant 6 right out front!This makes things easier to see.∫6xdx=∫6⋅1xdx \int \frac{6}{x} dx = \int 6 \cdot \frac{1}{x} dx ∫x6​dx=∫6⋅x1​dx

STEP 4

Why can we do this?Because it's a *constant* multiple!Remember, constants just scale things up or down, they don't change the *fundamental nature* of the function.

STEP 5

Now, we can **apply the constant multiple rule**.This rule says that the integral of a constant times a function is the same as the constant times the integral of the function.So, we get:∫6⋅1xdx=6⋅∫1xdx \int 6 \cdot \frac{1}{x} dx = 6 \cdot \int \frac{1}{x} dx ∫6⋅x1​dx=6⋅∫x1​dx

STEP 6

Think of it like this: if you're finding the area under a curve that's 6 times taller than another curve, the total area will be 6 times bigger!

STEP 7

STEP 8

STEP 9

6ln⁡∣x∣+C 6 \ln|x| + C 6ln∣x∣+C

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionFind the most general antiderivative by evaluating the following indefinite integral: $\int \frac{6}{x} d x=$ $\square$
NOTE: The general antiderivative should contain an arbitrary constant.

1. Rewrite the integral
2. Apply the constant multiple rule
3. Integrate

Alright, let's rewrite our integral to make it super clear what we're working with.
We can pull that constant 6 right out front!
This makes things easier to see.
$\int \frac{6}{x} dx = \int 6 \cdot \frac{1}{x} dx$

Why can we do this?
Because it's a constant multiple!
Remember, constants just scale things up or down, they don't change the fundamental nature of the function.

Now, we can apply the constant multiple rule.
This rule says that the integral of a constant times a function is the same as the constant times the integral of the function.
So, we get:
$\int 6 \cdot \frac{1}{x} dx = 6 \cdot \int \frac{1}{x} dx$

$6 \ln|x| + C$