QuestionFind the integral $\int \frac{x^{2}}{x^{3}+9} d x$ and provide the result in terms of $c$ .

Studdy Solution

STEP 1

Assumptions1. The integral we are trying to solve is $\int \frac{x^{}}{x^{3}+9} d x$ . The integral is indefinite, which means we are looking for an antiderivative of the function inside the integral.
3. The function inside the integral is a rational function, which means we can use the method of partial fractions to solve it.

STEP 2

We can rewrite the integral as follows $\int \frac{x^{2}}{x^{}+9} d x = \int \frac{x^{2}}{x^{}+9} d x$

STEP 3

We can see that the derivative of the denominator $x^{3}+9$ is $3x^{2}$ , which is a multiple of the numerator $x^{2}$ . This suggests that we can use a u-substitution to solve the integral.
Let $u = x^{3}+9$ . Then, $du =3x^{2} dx$ .

STEP 4

We can rewrite the integral in terms of $u$ and $du$ . However, we need to adjust for the fact that the derivative of $u$ is $3x^{2}$ , not $x^{2}$ . We can do this by multiplying the integral by $\frac{1}{3}$ .
$\int \frac{x^{2}}{x^{3}+9} d x = \frac{1}{3} \int \frac{du}{u}$

STEP 5

Now we can solve the integral. The integral of $\frac{1}{u}$ with respect to $u$ is $\ln |u|$ .
$\frac{1}{3} \int \frac{du}{u} = \frac{1}{3} \ln |u| + C$

STEP 6

Finally, we substitute $u = x^{3}+9$ back into the equation to get the final answer.
$\frac{1}{3} \ln |u| + C = \frac{1}{3} \ln |x^{3}+9| + C$ So, the solution to the integral $\int \frac{x^{2}}{x^{3}+9} d x$ is $\frac{1}{3} \ln |x^{3}+9| + C$ .

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