This video tutorial covers the integration of exponential functions, focusing on seven problems. It begins with an introduction to the basic formula for integrating exponential functions and the use of u-substitution. Each problem is solved step-by-step, demonstrating techniques such as expanding expressions, simplifying complex fractions, and handling tricky integrals. The tutorial emphasizes the importance of back-substitution and provides clear explanations for each solution.

What is the basic formula for integrating an exponential function?

• Integral of e^u du = u^e - C

• Integral of e^u du = e^u + C

• Integral of e^u du = u^e + C

• Integral of e^u du = e^u - C

In the first problem, what substitution is used to solve the integral of e^4x?

• u = 2x

• u = x^4

• u = x/4

• u = 4x

How is the integral of e^-2x rewritten before applying u-substitution?

• As e^-2x

• As e^-x

• As e^x

• As e^2x

What method is used to expand (e^(3x+2))^2 in the third problem?

• FOIL method

• Integration by parts

• Partial fraction decomposition

• Substitution method

In problem 4, how is the integral of e^x - 4e^-x over e^x simplified?

• By using partial fractions

• By combining terms

• By separating the numerator

• By multiplying by e^x

How is the square root of e^3x converted for integration in problem 5?

• As e^(x^3)

• As e^(3x)

• As e^(x/3)

• As e^(3x/2)

What technique is used to solve the integral of dx over 1 plus e^x in problem 6?

• Using partial fractions

• Multiplying by e^-x

• Using integration by parts

• Multiplying by e^x

In problem 7, what substitution is used to solve the integral of e^(x^3) * x^2?

• u = x^3

• u = x^3/3

• u = 3x^2

• u = x^2