Question

Make a substitution to express the integrand as a rational function and then evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) ex (ex − 4)(e2x + 1) dx

Answer 1

Answer:

(1/17) [In |eˣ - 4| - 0.5 In |e²ˣ + 1| - 4 arctan eˣ] + c

Step-by-step explanation:

The solution to the question is provided in the attached image.

The right substitution is y = eˣ

y = eˣ

(dy/dx) = eˣ

dy = eˣ dx

dx = (dy/eˣ)

And x = In y

We then obtain an expression that can be resolved into integrable form by resolving into partial fractions.

After this, the partial fractions is then integrated; giving the final answer obtained.

Answer 2

Answer:

The integral is

(1/34)[2ln|e^x - 4| - ln|e^(2x) + 1| - 8arctan(e^x)] + C

Step-by-step explanation:

A substitution u = e^x was made into the given function.

This made du = e^xdx, and dx = du/e^x = du/u.

The function is then in terms of u. This is then resolve into partial fractions, and the resulting resolution is integrated using the properties of integration.

CHECK ATTACHMENTS FOR THE WORKINGS.