Question

Could someone please help me with the following indefinite integral? (click to see)

Answer 1

\[2\int\limits_{}^{}\frac{ 1 }{ x }dx+3\int\limits_{}^{}e^xdx\]

Answer 2

\[=2\ln|x|+3e^x+c\]

Answer 3

Could someone please explain why the natural log of x must be an absolute value of x? I do not understand what it can't be a negative x-value.

Answer 4

Natural log is undefined for values less than zero. When x = 0, \[\ln (x)=- \infty\]But when x < 0, the answer is undefined.

Answer 5

Unless you fancy complex numbers :)
\[\ln(-\alpha) (\alpha > 0)=\ln( \alpha e^{i \pi})=\ln(\alpha)+i \pi\]

Answer 6

I figured that basic calculus wasn't at that level of math quite yet. ^-^