Integration substitution rule

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Homework Statement



mffundodmainsubrule.png


Homework Equations



below

The Attempt at a Solution



I have shown that the first identity holds true. Because this is true without it being surrounded by an integral I guess you would need to integrate it all around the same contour ##C##. So say I have:

## _C \int \frac{f'(\gamma(t)) d(\gamma(t))}{f(\gamma((t))} = _C \int \frac{f'(t) d(t)}{f(t)} + _C \int \frac{k c}{ct+d} dt ##

(Although I am confused because usually when you have what is the left-hand side you have the substitutin rule if ##f(x) \to f(y(x))## then ## _{c(x)} \int f(x) dx \to _{c(y(x))} \int f(y(x)) y'(x) dx## where ##c(x)## is the contour of integration in the original coordinates. (2).)

In which case then does the second expression via applying the substituion integral rule on the left hand side term (as in the first expression) to ## c \to \gamma t ## and ##f(\gamma t) \to f(t) ## etc. However substitution rule as above (2) I can't seem to understand, I have if ## c \to \gamma t ## , ##f'(\gamma t) \to f'(\gamma (\gamma(t)) ## etc to get ## _{\gamma(c)} \int \frac{f'(\gamma(\gamma(t)) d(\gamma(\gamma(t))}{f(\gamma(\gamma(t)))} ##

Have I applied the substitution rule wrong or should my starting expression instead be:

## _{\gamma(C)} \int \frac{f'(\gamma(t)) d(\gamma(t))}{f(\gamma((t))} = _C \int \frac{f'(t) d(t)}{f(t)} + _C \int \frac{k c}{ct+d} dt ##

many thanks
 

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