U-Substitution and Integration Techniques

This video tutorial focuses on evaluating definite integrals using u-substitution. It begins with an introduction to the concept and proceeds with three detailed examples. The first example involves integrating 2x(x^2+4)^2, the second example deals with 4x√(16-x^2), and the third example covers 2x(1+x^2)^3. Each example is solved step-by-step, highlighting the process of changing variables, adjusting limits, and finding antiderivatives. The tutorial emphasizes the importance of converting limits from x-values to u-values and provides a comprehensive understanding of u-substitution in calculus.

What is the first step in evaluating a definite integral using u-substitution?

• Set u equal to a part of the integrand.

• Find the derivative of the function.

• Evaluate the integral directly.

• Change the limits of integration.

When using u-substitution, what must be done after changing the variable from x to u?

• Multiply the integrand by the derivative of u.

• Find the derivative of the new function.

• Change the limits of integration to match the new variable.

• Evaluate the integral immediately.

In the first example, what is the new upper limit after substitution if the original upper limit was 2?

• 4

• 10

• 6

• 8

What is the derivative of 16 - x^2 in the second example?

• -x

• x

• -2x

• 2x

In the second example, what is the value of u when x is 0?

• 32

• 16

• 8

• 0

What is the antiderivative of u^(1/2)?

• u^(3/2) / (3/2)

• u^(3/2) * (2/3)

• u^(1/2) / 2

• u^(1/2) * 2

In the final example, what is the expression for du if u = 1 + x^2?

• 2x dx

• 2 dx

• x dx

• dx

What is the new lower limit when x is 1 in the final example?

• 1

• 4

• 2

• 3

How do you rewrite 1/u^3 for integration?

• 3u

• 1/u

• u^(-3)

• u^3

What is the final answer for the last example's integral?

• 256/3

• 21/200

• 64/3

• 448/3