Integral of 1 over e to the x

The video tutorial explains how to find the integral of 1 over e to the x using u-substitution. It begins by rewriting the integral, then applies the u-substitution method to simplify the problem. The tutorial continues by solving the integral using antiderivatives and concludes with the final solution, which is negative e to the negative x plus a constant.

What is the first step in solving the integral of 1 over e to the x?

• Move e to the x to the numerator

• Differentiate the function

• Apply integration by parts

• Use partial fraction decomposition

What substitution is made for u in the u-substitution method?

• u = e^x

• u = -x

• u = 1/e^x

• u = x

What is the derivative of u when u = -x?

• 1

• -1

• x

• -x

How is dx expressed in terms of du after substitution?

• dx = e^u du

• dx = -du

• dx = du

• dx = u du

What is the integral of e to the u with respect to u?

• u e^u + C

• e^u + C

• ln|u| + C

• 1/u + C

What is the final expression for the antiderivative of 1 over e to the x?

• e^x + C

• -e^x + C

• -e^(-x) + C

• ln|x| + C

What constant is added to the antiderivative?

• e

• C

• 1

• 0

What is the purpose of replacing u with -x in the final step?

• To solve for x

• To revert to the original variable

• To simplify the expression

• To find the derivative