Math / Calculus

QuestionFind the indefinite integral. (Remember the constant of integration.) $\int \frac{1}{3 x^{4}} d x$ $\square$
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Studdy Solution

STEP 1

What is this asking? We need to find the indefinite integral of $\frac{1}{3x^4}$ , which means finding a function whose derivative is $\frac{1}{3x^4}$ , and don't forget to add the constant of integration! Watch out! Remember the power rule for integrals, and be careful with that fraction and negative exponents!

STEP 2

1. Rewrite the integrand
2. Apply the power rule
3. Simplify

STEP 3

Let's rewrite our integral to make it easier to work with.
We can pull the constant factor of $\frac{1}{3}$ out front.
Remember, constants can be factored out of integrals.
So, we have: $\int \frac{1}{3x^4} \, dx = \frac{1}{3} \int \frac{1}{x^4} \, dx$ Why did we do this?
It makes applying the power rule much cleaner!

STEP 4

Now, let's rewrite $\frac{1}{x^4}$ with a negative exponent.
Remember, $\frac{1}{x^n}$ is the same as $x^{-n}$ .
This is crucial for using the power rule for integration. $\frac{1}{3} \int \frac{1}{x^4} \, dx = \frac{1}{3} \int x^{-4} \, dx$ This sets us up perfectly for the next step!

STEP 5

Time to use the power rule for integration!
The power rule says: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ , where $n \neq -1$ .
In our case, $n$ is -4.
So, let's apply the power rule: $\frac{1}{3} \int x^{-4} \, dx = \frac{1}{3} \cdot \frac{x^{-4+1}}{-4+1} + C$

STEP 6

Let's simplify the exponent and the denominator: $\frac{1}{3} \cdot \frac{x^{-4+1}}{-4+1} + C = \frac{1}{3} \cdot \frac{x^{-3}}{-3} + C$ Almost there!

STEP 7

Let's multiply the fractions and rewrite the negative exponent: $\frac{1}{3} \cdot \frac{x^{-3}}{-3} + C = \frac{x^{-3}}{-9} + C = -\frac{1}{9x^3} + C$ And there we have it!
Don't forget that constant of integration, $C$ , which is super important for indefinite integrals.

STEP 8

The indefinite integral of $\frac{1}{3x^4}$ is $-\frac{1}{9x^3} + C$ .

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QuestionFind the indefinite integral. (Remember the constant of integration.) $\int \frac{1}{3 x^{4}} d x$ $\square$
Need Help? Read It $\square$ Master It $\square$ Submit Answer

Studdy Solution

STEP 1

STEP 2

1. Rewrite the integrand
2. Apply the power rule
3. Simplify

STEP 3

STEP 4

Now, let's rewrite $\frac{1}{x^4}$ with a negative exponent.
Remember, $\frac{1}{x^n}$ is the same as $x^{-n}$ .
This is crucial for using the power rule for integration. $\frac{1}{3} \int \frac{1}{x^4} \, dx = \frac{1}{3} \int x^{-4} \, dx$ This sets us up perfectly for the next step!

STEP 5

Time to use the power rule for integration!
The power rule says: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ , where $n \neq -1$ .
In our case, $n$ is -4.
So, let's apply the power rule: $\frac{1}{3} \int x^{-4} \, dx = \frac{1}{3} \cdot \frac{x^{-4+1}}{-4+1} + C$

STEP 6

Let's simplify the exponent and the denominator: $\frac{1}{3} \cdot \frac{x^{-4+1}}{-4+1} + C = \frac{1}{3} \cdot \frac{x^{-3}}{-3} + C$ Almost there!

STEP 7

Let's multiply the fractions and rewrite the negative exponent: $\frac{1}{3} \cdot \frac{x^{-3}}{-3} + C = \frac{x^{-3}}{-9} + C = -\frac{1}{9x^3} + C$ And there we have it!
Don't forget that constant of integration, $C$ , which is super important for indefinite integrals.

STEP 8

The indefinite integral of $\frac{1}{3x^4}$ is $-\frac{1}{9x^3} + C$ .

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QuestionFind the indefinite integral. (Remember the constant of integration.) ∫13x4dx\int \frac{1}{3 x^{4}} d x∫3x41​dx □\square□Need Help? Read It □\square□ Master It □\square□ Submit Answer

Studdy Solution

STEP 1

STEP 2

1. Rewrite the integrand2. Apply the power rule3. Simplify

STEP 3

STEP 4

STEP 5

STEP 6

Let's **simplify** the exponent and the denominator: 13⋅x−4+1−4+1+C=13⋅x−3−3+C \frac{1}{3} \cdot \frac{x^{-4+1}}{-4+1} + C = \frac{1}{3} \cdot \frac{x^{-3}}{-3} + C 31​⋅−4+1x−4+1​+C=31​⋅−3x−3​+C Almost there!

STEP 7

STEP 8

The indefinite integral of 13x4\frac{1}{3x^4}3x41​ is −19x3+C-\frac{1}{9x^3} + C−9x31​+C.

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QuestionFind the indefinite integral. (Remember the constant of integration.) $\int \frac{1}{3 x^{4}} d x$ $\square$
Need Help? Read It $\square$ Master It $\square$ Submit Answer

1. Rewrite the integrand
2. Apply the power rule
3. Simplify

Let's simplify the exponent and the denominator: $\frac{1}{3} \cdot \frac{x^{-4+1}}{-4+1} + C = \frac{1}{3} \cdot \frac{x^{-3}}{-3} + C$ Almost there!

The indefinite integral of $\frac{1}{3x^4}$ is $-\frac{1}{9x^3} + C$ .