Math / Calculus

QuestionFind the integral given below. Check your answer by differentiation. $\int \frac{6 x^{5}}{x^{6}+1} d x=$ $\square$ $+C$

Studdy Solution

STEP 1

What is this asking? We need to find the integral of a fraction with $x$ and then double-check our work by taking the derivative of our answer. Watch out! Don't forget the $+C$ when integrating and remember the chain rule when differentiating!

STEP 2

1. Set up the substitution
2. Compute the integral
3. Check the result

STEP 3

Let's define $u = x^6 + 1$ .
This will help us simplify the integral.
We're picking this specific $u$ because we see an $x^6$ in the numerator, which is very close to the derivative of the denominator $x^6 + 1$ .

STEP 4

Now, let's find the derivative of $u$ with respect to $x$ .
We get $\frac{du}{dx} = 6x^5.$

STEP 5

Next, we solve for $dx$ .
We get $dx = \frac{du}{6x^5}.$ This helps us rewrite the integral in terms of $u$ .

STEP 6

Let's rewrite the original integral using our substitution: $\int \frac{6x^5}{x^6 + 1} dx = \int \frac{6x^5}{u} \cdot \frac{du}{6x^5}.$

STEP 7

Now, we can simplify the integral by dividing to one.
Notice that the $6x^5$ terms divide to one, leaving us with $\int \frac{1}{u} du.$ Much nicer, right?!

STEP 8

We know that the integral of $\frac{1}{u}$ is $\ln|u|$ .
So, we have $\int \frac{1}{u} du = \ln|u| + C.$

STEP 9

Finally, let's substitute back $x^6 + 1$ for $u$ to get our answer in terms of $x$ : $\ln|x^6 + 1| + C.$ Since $x^6$ is always non-negative and we add $1$ to it, $x^6 + 1$ is always positive.
Therefore, we can remove the absolute value: $\ln(x^6 + 1) + C.$

STEP 10

To check our work, let's differentiate our answer $\ln(x^6 + 1) + C$ using the chain rule.

STEP 11

The derivative of $\ln(x^6 + 1)$ is $\frac{1}{x^6 + 1}$ multiplied by the derivative of the inside function $x^6 + 1$ , which is $6x^5$ .
And, of course, the derivative of the constant $C$ is $0$ .

STEP 12

Putting it all together, we get $\frac{d}{dx} (\ln(x^6 + 1) + C) = \frac{1}{x^6 + 1} \cdot 6x^5 + 0 = \frac{6x^5}{x^6 + 1}.$

STEP 13

This matches our original integrand, so we know our integration is correct!

STEP 14

$\int \frac{6 x^{5}}{x^{6}+1} d x = \ln(x^6 + 1) + C$

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