Math  /  Calculus

QuestionDetermine the indefinite integral 36x5x6+5dx\int \frac{36 x^{5}}{x^{6}+5} d x by substitution. (It is recommended that you check your results by differentiation.) Use capital C for the free constant.
Answer: \square
Hint: Follow Example 6.

Studdy Solution

STEP 1

What is this asking? We need to find the indefinite integral of a function that looks kinda scary, but don't worry, substitution will save the day! Watch out! Don't forget the "+ C" at the end – it's like the cherry on top of our integration sundae!
Also, remember to double-check your work by differentiating your answer; it's like hitting the "undo" button to make sure we get back to where we started.

STEP 2

1. Set up the substitution
2. Perform the substitution
3. Integrate
4. Substitute back
5. Check the result

STEP 3

Let's **define** our *u* substitution.
We're going to let u=x6+5u = x^6 + 5, because its derivative, 6x56x^5, is hiding in the numerator.
This is like finding a secret passage in a math dungeon!

STEP 4

Now, we **find** the derivative of *u* with respect to *x*.
So, dudx=6x5\frac{du}{dx} = 6x^5.

STEP 5

Next, we **solve** for *dx*.
We can rewrite the previous equation as du=6x5dxdu = 6x^5 dx.
Then, we divide both sides by 6x56x^5 to get dx=du6x5dx = \frac{du}{6x^5}.
This is like getting the key to unlock the next part of the problem!

STEP 6

Let's **rewrite** our original integral using our substitution: 36x5x6+5dx=36x5udu6x5\int \frac{36x^5}{x^6 + 5} dx = \int \frac{36x^5}{u} \cdot \frac{du}{6x^5}.

STEP 7

Now, **simplify**!
Notice that the x5x^5 terms divide to one, and 3636 divided by 66 is **6**.
So, we're left with 6udu\int \frac{6}{u} du.
Much cleaner, right?

STEP 8

We know the integral of 1u\frac{1}{u} is lnu\ln|u|.
So, the integral of 6u\frac{6}{u} is just 6lnu6 \ln|u|.
Don't forget the **constant of integration**, *C*!
So we have 6lnu+C6 \ln|u| + C.

STEP 9

Remember, *u* was just a stand-in for x6+5x^6 + 5.
Let's **put it back** in: 6lnx6+5+C6 \ln|x^6 + 5| + C.
Since x6x^6 is always non-negative, and we're adding **5**, we can drop the absolute value signs without changing the answer, giving us 6ln(x6+5)+C6 \ln(x^6 + 5) + C.

STEP 10

Let's **differentiate** our answer to make sure we get back to the original integrand.
The derivative of 6ln(x6+5)+C6 \ln(x^6 + 5) + C is 61x6+56x56 \cdot \frac{1}{x^6 + 5} \cdot 6x^5, which simplifies to 36x5x6+5\frac{36x^5}{x^6 + 5}.
Boom! We're back where we started.

STEP 11

Our **final answer** is 6ln(x6+5)+C6 \ln(x^6 + 5) + C.

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