Questions: Find the antiderivative: [ int frac8 x^6+6 x^3+8x^5 d x= ] [ +C ]

Find the antiderivative: [ int frac8 x^6+6 x^3+8x^5 d x= ] [ +C ]
Transcript text: Find the antiderivative: \[ \int \frac{8 x^{6}+6 x^{3}+8}{x^{5}} d x= \] \[ +C \]
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Solution

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Solution Steps

To find the antiderivative of the given expression, we first simplify the integrand by dividing each term in the numerator by the term in the denominator. This will give us a polynomial expression. Then, we integrate each term of the polynomial separately using the power rule for integration.

Step 1: Simplifying the Integrand

We start with the integral

\[ \int \frac{8 x^{6}+6 x^{3}+8}{x^{5}} \, dx. \]

We simplify the integrand by dividing each term in the numerator by \(x^5\):

\[ \frac{8 x^{6}}{x^{5}} + \frac{6 x^{3}}{x^{5}} + \frac{8}{x^{5}} = 8x + 6x^{-2} + 8x^{-5}. \]

Step 2: Integrating Each Term

Next, we integrate each term separately:

  1. The integral of \(8x\) is

\[ \int 8x \, dx = 4x^2. \]

  1. The integral of \(6x^{-2}\) is

\[ \int 6x^{-2} \, dx = -6x^{-1} = -\frac{6}{x}. \]

  1. The integral of \(8x^{-5}\) is

\[ \int 8x^{-5} \, dx = -\frac{8}{4x^4} = -\frac{2}{x^4}. \]

Step 3: Combining the Results

Combining all the results from the integrations, we have:

\[ \int \frac{8 x^{6}+6 x^{3}+8}{x^{5}} \, dx = 4x^2 - \frac{6}{x} - \frac{2}{x^4} + C, \]

where \(C\) is the constant of integration.

Final Answer

Thus, the antiderivative is

\[ \boxed{4x^2 - \frac{6}{x} - \frac{2}{x^4} + C}. \]

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