Evaluate the following integrals :
`int(a/x^(2)+b/x)dx`
(a and b are constant)

Answer

To evaluate the integral \( \int \left( \frac{a}{x^2} + \frac{b}{x} \right) dx \), we can break it down into two separate integrals. Let's go through the steps one by one.

### Step 1: Split the Integral
We can rewrite the integral as:
\[ \int \left( \frac{a}{x^2} + \frac{b}{x} \right) dx = \int \frac{a}{x^2} dx + \int \frac{b}{x} dx \]

### Step 2: Evaluate the First Integral
The first integral is \( \int \frac{a}{x^2} dx \). We can rewrite \( \frac{a}{x^2} \) as \( a x^{-2} \):
\[ \int a x^{-2} dx \]
Using the power rule for integration, which states that \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \) (where \( n \neq -1 \)), we have:
\[ \int a x^{-2} dx = a \cdot \frac{x^{-1}}{-1} = -\frac{a}{x} \]

### Step 3: Evaluate the Second Integral
Now, we evaluate the second integral \( \int \frac{b}{x} dx \):
\[ \int b \cdot x^{-1} dx \]
The integral of \( \frac{1}{x} \) is \( \ln |x| \), so:
\[ \int b \cdot x^{-1} dx = b \ln |x| \]

### Step 4: Combine the Results
Now we combine the results from the two integrals:
\[ \int \left( \frac{a}{x^2} + \frac{b}{x} \right) dx = -\frac{a}{x} + b \ln |x| + C \]
where \( C \) is the constant of integration.

### Final Result
Thus, the final result of the integral is:
\[ -\frac{a}{x} + b \ln |x| + C \]

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