Question

Question #b261d

Answer 1

Here are two descriptions of calculating the integral.

Explanation:

Finding the `r` and `k` requested

The expression `r[((x+k)/sqrtr)^2 +1]` can be expanded to

`r((x+k)^2/r) + r = (x+k)^2+r`

So another way of describing what we've been asked to do is to complete the square in the denominator.

We want

`x^2+8x+20 = x^2+2kx+k^2+r`.

So `k = 4` and `r = 4`.

Now we have

`int 1/(x^2+8x+20) dx = int 1/(4(((x+4)/2)^2+1)) dx`

`= 1/4 int 1/(((x+4)/2)^2+1) dx`.

We can integrate by using the subsitution `u = (x+4)/2` to get

`= 1/2 int 1/(u^2+1) du = 1/2 tan^-1u +C`

and finish with

`1/2 tan^-1 ((x+4)/2) +C`

OR

The method and details above are new to me. Here's how I would write the solution.

To find `int 1/(x^2+8x+20) dx` we hope that we can use a substitution to turn the quadratic into `u^2+1`.

To attempt to do this, complete the square.

`x^2+8x + color(white)"ssssssss" +20`

`1/2 * 8 = 4` `" and "` `4^2 = 16`, so add and subtract 16 in the space above.

`x^2+8x +16 -16 + 20`. Now simplify and factor

`(x+4)^2+4`

To make this `u^2+1` we need to factor out the `4`.

`4((x+4)^2/4 + 1)`

`int 1/(x^2+8x+20) dx = 1/4 int 1/(((x+4)^2/4 + 1)) dx`

With the same choice of `u` as above, we finish the same way as above.

Perhaps the differences are subtle, but I learned the second reasoning and didn't recognize what was being requested in `r` and `k`.