When do you use In or log in Integrals?

[pillar]

1.For example

a.http://img40.imageshack.us/img40/9514/problem2m.png

b.http://img193.imageshack.us/img193/4250/problem2ans.png When do you use ln in integrals?

 

[HallsofIvy]

One of the things you should have learned early is that
\int \frac{1}{x}dx= ln|x|+ C

\int sin(x) dx is, as you first say, - cos(x)+ C. It is certainly NOT equal to "ln|cos(x)|+ C. I don't know why you would even suggest that.

 

[pillar]

One of the things you should have learned early is that
\int \frac{1}{x}dx= ln|x|+ C

\int sin(x) dx is, as you first say, - cos(x)+ C. It is certainly NOT equal to "ln|cos(x)|+ C. I don't know why you would even suggest that.

Sorry ln|cos(x)|+ C+ must be the integral of tan(x) then.

 

[HallsofIvy]

Sorry ln|cos(x)|+ C+ must be the integral of tan(x) then.

Almost. If you let u= cos(x) then du= -sin(x)dx so
\int tan(x)dx= \int\frac{sin(x)}{cos(x)}dx= -\int \frac{du}{u}=-ln|u|+ C= -ln|cos(x)|+ C

 

[Mark44]

BTW, the shorthand form is not In--it is ln, an abbreviation for logarithmus naturalis, Latin for natural logarithm.

 

[njama]

Just find the derivative of ln|cos(x)| and you will be sure, what the correct integral is.

[-ln|cos(x)| ]'=(-ln(u))'u'=(-1/cos(x))*(-sin(x))=sin(x)/cos(x)=tan(x)

So \int{tan(x)}=-ln|cos(x)|+C.