`int e^x((x^2+1))/((x+1)^2)dx is equal to` `((x-1)/(x+1))e^x+c` (b) `e^x((x+1)/(x-1))+c` `e^x(x+1)(x-1)+c` (d) none of these

Options

A. `((x-1)/(x+1))e^(x)+c`
B. `e^(x)((x+1)/(x-1))+c`
C. `e^(x)(x+1)(x-1)+c`
D. none of these

Explanation

To solve the integral \( \int e^x \frac{x^2 + 1}{(x + 1)^2} \, dx \), we can follow these steps:

### Step 1: Rewrite the integrand
We start by rewriting the integrand:
\[ \frac{x^2 + 1}{(x + 1)^2} = \frac{x^2 - 1 + 2}{(x + 1)^2} = \frac{(x^2 - 1) + 2}{(x + 1)^2} \]
This can be expressed as:
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