integration analysis

Consider the integral

[ sqrt(e^y) / e^y ] dy

This is an indefinite integral, so we need to find the most general antiderivative of the integrand f(y).

To find an antiderivative of f(y), we go through our list of integration methods:

Recognize elementary antiderivatives
Rewrite the integrand to make it easier
Use substitution to reverse the chain rule or simplify the integrand
Use integration by parts
Use inspection to see the value of a definite integral

to find one that works for this integrand. In this case, only one method is appropriate. We can rewrite the integrand: use properties of exponents to combine numerator and denominator of the fraction to give an expression that is more easily integrable. This is shown below.

Rewriting:
Use properties of exponents to combine numerator and denominator of the fraction to rewrite the integral:

[ sqrt(e^y) / e^y ] = e^-y/2.

Thus,

[ sqrt(e^y) / e^y ] dy =

e^-y/2dy

which can be evaluated using substitution (of "ax"), to obtain

e^-y/2dy = (-2 e^-y/2) + C.

Explanation for rewritten term(s)
Substitution (of "ax")
Let w = -1/2 y. Then w' = -1/2, so (1/(-1/2)) dw = dy. The integral can therefore be rewritten as

e^-y/2 dy =

e^w (1/(-1/2)) dw = -2 e^w + C

Thus, substituting back for w,

e^-y/2 dy = -2 e^-y/2 + C

integration tutorial: analysis