Question

integrate (pi)dx / sqrt (pi)x^2

Answer 1

\[\frac{ \Pi dx }{ \sqrt{\Pi x^{2}} }\]

Answer 2

note that since \(\pi/\sqrt{\pi}=\sqrt\pi\) is a constant it passes right thru:$$\int\frac{\pi}{\sqrt{\pi} x^2}dx=\int\frac{\sqrt\pi}{x^2}dx=\sqrt{\pi}\int\frac1{x^2}dx$$

Answer 3

now observe \(1/x^2=x^{-2}\) and use the power rule:
$$\int x^ndx=\frac1{n+1}x^{n+1}+C$$

Answer 4

sqrt (pi x^2)

Answer 5

x^2 is inside the sqrt. :)
so x^2 becomes x

Answer 6

sqrt(pi) * ln(x) + C

Answer 7

okay, not a worry: $$\sqrt{\pi x^2}=\sqrt{\pi}\sqrt{x^2}=\sqrt{\pi}|x|$$so the constant is the same but now we integrate \(1/|x|\) rather than \(1/x^2\). this is a little trickier. if you're told that \(x\ge0\) then this is just \(1/x\) of course

Answer 8

$$\sqrt\pi\int\frac1{|x|}dx=\sqrt\pi\int\frac1xdx=\sqrt{\pi}\log |x|+C$$

Answer 9

for \(x\ge 0\)

Answer 10

logarithm does not exist for negative numbers. Also, the original function has an x in the denominator and therefore x = 0 is not in the domain of the function.
So sqrt(pi) * ln(x) + C for x > 0.

Answer 11

just one question

Answer 12

why is it many uses log and ln interchangably

Answer 13

? :)

Answer 14

In calculus natural logarithm is what is meant when we say logarithm unless the problem explicitly says what the base is.

Answer 15

natural logarithm is well natural in the sense it is the logarithmic function that gives the area bounded by the positive branch of \(1/x\)

Answer 16

and anyways @ranga is correct that this particular function 1/x is not integrable over any non-trivial interval containing 0 but that's not in general true for other functions. take for example:$$\int_{-1}^2\frac1{x^{1/2}}dx$$even though \(1/x^{1/2}\) is undefined at \(x=0\) the function is still integrable over that piece. although, yes, you are correct that \(\log x\) is undefined at \(x=0\)

Answer 17

engineers use \(\ln\) because they are filthy

Answer 18

ok. thanks guyz!! :D

Answer 19

just for the note, :)
i'm an engineer. ;)

Answer 20

\[\frac{ d }{ dx}\ln(x) = \frac{ 1 }{ x }\quad but \quad \frac{ d }{ dx}\log_{10} (x) = \frac{ 1 }{ \ln10 } \times \frac{ 1 }{ x }\]

Answer 21

i mean.. not yet an engineer but becoming an engineer. :)