Question

For `int (x - 1)/(x - 1)^3 e^ x dx = e^x * f(x) + c`, where f(x) = (x + 1)².

Answer 1

This statement is false.

Explanation:

1. Simplify the Integrand

The given expression in the integral is `(x - 1)/(x - 1)^3`. This simplifies by canceling one term of (x – 1) from the numerator and denominator:

`(x - 1)/(x - 1)^3 e^x = 1/(x - 1)^2 e^x`

2. Evaluate the Integral

We are looking for `int e^x/(x - 1)^2 dx`. To check if the result is e^x · f(x) + c where f(x) = (x + 1)², we can differentiate the proposed solution:

`d/dx [e^x (x + 1)^2]`

Using the product rule:

`d/dx [e^x (x + 1)^2] = e^x (x + 1)^2 + e^x * 2(x + 1)`

Factoring out e^x:

e^x[(x + 1)² + 2x + 2]

= e^x[x² + 2x + 1 + 2x + 2]

= e^x(x² + 4x + 3)

3. Compare Results

The derivative of the proposed answer, e^x(x² + 4x + 3), does not match our simplified integrand `e^x/(x - 1)^2`.  Furthermore, standard integration of `e^x/(x - 1)^2` typically involves terms like `(-e^x)/(x - 1)` when using the form `int e^x[f(x) + f^'(x)] dx`, but it does not result in (x + 1)².