integration analysis

Consider the integral

[ (e^4x + 1) / e^2x ] dx

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

To find an antiderivative of f(x), 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: split the integrand into two parts and use properties of exponents to combine the numerator and denominators of the resulting ratios of exponentials to give an expression that is more easily integrable. This is shown below.

Rewriting:
Split the integrand into two parts and use properties of exponents to combine the numerator and denominators of the resulting ratios of exponentials to rewrite the integral:

[ (e^4x + 1) / e^2x ] = e^2x + e^-2x.

This can be broken into separate integrals,

[ (e^4x + 1) / e^2x ] dx =

e^2xdx+

e^-2xdx

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

e^2xdx+

e^-2xdx = ((1/2) e^2x) + (-(1/2) e^-2x) + C.

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

e^2x dx =

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

Thus, substituting back for w,

e^2x dx = (1/2) e^2x + C

Substitution (of "ax")
Let w = -2 x. Then w' = -2, so (1/(-2)) dw = dx. The integral can therefore be rewritten as

e^-2x dx =

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

Thus, substituting back for w,

e^-2x dx = -(1/2) e^-2x + C

integration tutorial: analysis