How to integrate e^(5x) between the limits 0 and 1.

Note that by the chain rule if the function y is such that y(x)=f(g(x)), where f and g are functions, then the derivative of y wrt x is given by

dy/dx = (df/dg)*(dg/dx).

Hence if we let the function y be e^(5x) and g(x)=5x then y(x)=e^(g(x)). Then by the chain rule as detailed above dy/dx = 5*e^(5x).

Note that this is similar to the function we're integrating e^(5x). In fact the derivative of (1/5)*e^(5x) is e^(5x). Let F(x) be this function.

Hence the value of the integral between the limits 0 and 1 is the difference of this function evaluated at the limits, that is F(1)-F(0) which is (1/5)*(e^(5)-1).