U-Substitution in Integral Calculus

This lesson covers evaluating indefinite integrals using the method of u-substitution. It provides two examples: the first involves substituting u for 9 minus e to the x, and the second uses natural log x. The video explains how to determine u, calculate du, and rewrite the integrals in terms of u. It also demonstrates applying the power rule of integration and simplifying the results.

What is the primary goal of using u-substitution in evaluating indefinite integrals?

• To determine the limits of integration

• To solve differential equations

• To find the derivative of the function

• To simplify the integrand by changing variables

In the first example, what is chosen as the substitution variable u?

• e to the power of x

• 9 minus e to the power of x

• x squared

• Natural log of x

How is du determined in the first example?

• By adding a constant

• By differentiating 9 minus e to the power of x

• By multiplying by a constant

• By integrating the function

What is the next step after substituting u and du in the first example?

• Differentiate the function

• Find the limits of integration

• Rewrite the integral in terms of u

• Apply the chain rule

What rule is applied to integrate the function in terms of u in the first example?

• Product rule

• Chain rule

• Quotient rule

• Power rule

How is the final result of the first example expressed?

• In terms of x

• As a differential equation

• As a definite integral

• In terms of u

In the second example, what is chosen as the substitution variable u?

• x squared

• Natural log of x

• e to the power of x

• 9 minus e to the power of x

What is the derivative of natural log x used for in the second example?

• To find the limits of integration

• To simplify the integrand

• To determine du

• To solve a differential equation

How is the denominator simplified in the second example?

• By differentiating

• By multiplying by a constant

• By adding a constant

• By factoring out a constant

What is the final form of the antiderivative in the second example?

• As a definite integral

• As a differential equation

• In terms of x

• In terms of u