QuestionEvaluate the integral: $\int \frac{6 y^{8}+12 y^{2}-y^{6}}{3 y^{7}} d y$

Studdy Solution

STEP 1

Assumptions1. We are given the integral $\int \frac{6 y^{8}+12 y^{}-y^{6}}{3 y^{7}} d y$ . We know that the integral of a sum or difference of functions is the sum or difference of their integrals.
3. We know that the integral of a constant times a function is the constant times the integral of the function.

STEP 2

First, we can simplify the integrand by dividing each term in the numerator by $y^{7}$ .
$\int \frac{6 y^{8}+12 y^{2}-y^{6}}{ y^{7}} d y = \int \frac{6 y^{8}}{ y^{7}} d y + \int \frac{12 y^{2}}{ y^{7}} d y - \int \frac{y^{6}}{ y^{7}} d y$

STEP 3

implify each term in the integral.
$= \int2 y d y + \int y^{-5} d y - \int \frac{1}{3 y} d y$

STEP 4

Now, we can integrate each term separately.The integral of $2y$ with respect to $y$ is $y^{2}$ , the integral of $4y^{-}$ with respect to $y$ is $-\frac{4}{4y^{-4}}$ , and the integral of $\frac{1}{3y}$ with respect to $y$ is $\frac{1}{3} \ln |y|$ .
$= y^{2} - y^{-4} - \frac{1}{3} \ln |y| + C$ Where $C$ is the constant of integration.

STEP 5

Finally, we can simplify the expression by getting rid of the negative exponent.
$= y^{2} - \frac{1}{y^{4}} - \frac{1}{3} \ln |y| + C$ This is the antiderivative of the given function.

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