`int(ax^(-2)+bx^(-1)+c)/(x^(-3))dx=`

Options

A. `2ax^(2)+3bx^(3)+4cx^(4)+c_(1)`
B. `6ax^(2)+4bx^(3)+3cx^(4)+c_(1)`
C. `a+b+cx^(-2)+c_(1)`
D. `(ax^(2))/(2)+(bx^(3))/(3)+(cx^(4))/(4)+c_(1)`

Answer

To solve the integral

\[ \int \frac{ax^{-2} + bx^{-1} + c}{x^{-3}} \, dx, \]

we can start by simplifying the integrand.

### Step 1: Rewrite the integrand
We can rewrite the expression by dividing each term in the numerator by \(x^{-3}\):

\[ \frac{ax^{-2}}{x^{-3}} + \frac{bx^{-1}}{x^{-3}} + \frac{c}{x^{-3}}. \]

This simplifies to:

\[ ax^{1} + bx^{2} + cx^{3}. \]

### Step 2: Set up the integral
Now we can rewrite the integral as:

\[ \int (ax + bx^2 + cx^3) \, dx. \]

### Step 3: Integrate term by term
We can integrate each term separately:

1. The integral of \(ax\) is \(\frac{a}{2} x^2\).
2. The integral of \(bx^2\) is \(\frac{b}{3} x^3\).
3. The integral of \(cx^3\) is \(\frac{c}{4} x^4\).

Putting it all together, we have:

\[ \int (ax + bx^2 + cx^3) \, dx = \frac{a}{2} x^2 + \frac{b}{3} x^3 + \frac{c}{4} x^4 + C, \]

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

### Final Result
Thus, the final result of the integral is:

\[ \frac{a}{2} x^2 + \frac{b}{3} x^3 + \frac{c}{4} x^4 + C. \]

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