Question

How do you evaluate the definite integral `int (x^2+2x)/(x+1)^2` from `[0, 1]`?

Answer 1

`1/2`

Explanation:

Before evaluating, we always have to find the integral. We look to simplify the integral as much as possible to see if anything can be cancelled.

`int_0^1 (x(x + 2))/(x + 1)^2`

We can't cancel anything, but a u-substitution would be effective. Let `u = x + 1`. Then `du = dx`. Also note that `x = u - 1` and `x + 2 = u + 1`. The integral therefore becomes:

`int_0^1 ((u - 1)(u + 1))/u^2 du`

Expand this:

`int_0^1 (u^2 - u + u - 1)/u^2 du`

`int_0^1 (u^2 - 1)/u^2 du`

Break into separate fractions.

`int_0^1 u^2/u^2 - 1/u^2 du`

`int_0^1 1 - 1/u^2 du`

`int_0^1 1 - u^-2 du`

You can now integrate this as `intx^ndx = x^(n + 1)/(n + 1) + C`, where `n != -1`.

`[u + 1/u]_0^1`

Reverse the substitution, since the initial variable wasn't `u`, it was `x`.

`[x + 1 + 1/(x + 1)]_0^1`

Evaluate using the second fundamental theorem of calculus, which states that `int_a^b F(x)dx = f(b) - f(a)`, where `f'(x) = F(x)` in all `[a, b]`.

`1 + 1 + 1/(1 + 1) - (0 + 1 + 1/(0 + 1))`

`2 + 1/2 - 2`

`1/2`

Hopefully this helps!