I get everything, but I can't seem to crack this one out, only one I got problems on..
-sqrt[x^2 +1]/x + ln(x + sqrt[x^2 + 1])
for simplicity, let u=sqrt(x^2+1)
-u/x+ ln(x+u)
the derivative is
u/x^2 -1/u du/dx + 1/(x+u) * (1+du/dx)
then, du/dx=1/2 (x^2+1)^-.5 (2x) or
x/sqrt(x^2+1) check me.
I slugged it out the long way without bobpursley's substitution, and got a final of
√(x^2 + 1) / x^2
It would be a horrible mess to type, do you have an answer from the back of the book?
To simplify the expression (-sqrt[x^2 + 1]/x) + ln(x + sqrt[x^2 + 1]), we can take it step by step.
1. First, let's simplify the square root term. The square root of (x^2 + 1) can be expressed as (x + sqrt[x^2 + 1]) * (x - sqrt[x^2 + 1]). So, we can rewrite the numerator as -sqrt[x^2 + 1] / x = - (x - sqrt[x^2 + 1]) / x.
2. Next, observe that the denominator x cancels out with (x - sqrt[x^2 + 1]) in the numerator, leaving -1 + sqrt[x^2 + 1] in the numerator. Now, we have (-1 + sqrt[x^2 + 1]) / x.
3. Finally, we can express the natural logarithm term as ln(x + sqrt[x^2 + 1]) = ln[(x + sqrt[x^2 + 1]) / x].
Combining the simplified numerator and the logarithm term, we get (-1 + sqrt[x^2 + 1]) / x + ln[(x + sqrt[x^2 + 1]) / x].