Question

What is `int ln (x^2+14x+24)dx`?

Answer 1

`intln(x^2 + 14x + 24)dx`
`=x(ln(x^2 + 14x + 24)-2)+12ln(x+12)+2ln(x+2) + C`

Explanation:

For this problem, we will use the following:

• `int(f(x) + g(x))dx = intf(x)dx + int(g(x)dx`

• `int(ln(x)dx = x(ln(x)-1)+C`

• `ln(ab) = ln(a) + ln(b)`

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`intln(x^2 + 14x + 24)dx = intln((x + 12)(x+2))dx`

`=int(ln(x+12)+ln(x+2))dx`

`=intln(x+12)dx + intln(x+2)dx`

`=(x+12)(ln(x+12) - 1) + (x+2)(ln(x+2)-1) + C`

`=x(ln(x+12)+ln(x+2))+12ln(x+12)+2ln(x+2) - (x+12)-(x+2) + C`

`=xln((x+12)(x+2))+12ln(x+12)+2ln(x+2) -2x - 14 + C`

(We can combine the `-14` with `C` as `C` is arbitrary)

`=x(ln(x^2 + 14x + 24)-2)+12ln(x+12)+2ln(x+2) + C`