Question

State whether the following statement is true or false:

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:

`int(x - 1)/(x - 1)^3`

The integral is typically solved by splitting the numerator to match the denominator. Assuming the denominator is (x + 1)³

`int e^x  (x - 1)/(x + 1)^3 dx`

= `int e^x [((x + 1) - 2)/((x + 1)^3)] dx`

= `e^x [1/(x + 1)^2 - 2/(x + 1)^3]dx`

Using the standard formula:

∫ e^x [g(x) + g'(x)]dx = e^xg(x) + c

Let g(x) = `1/(x + 1)^2`

Then g'(x) = `-2/(x + 1)^3`

`e^x. 1/(x + 1)^2 + c`

The correct function is f(x) = `1/(x + 1)^2`

So the statement is false.