Math Snap

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.

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