Integration Techniques and Substitution

The video tutorial addresses a common question about evaluating definite integrals using the substitution method: whether new limits of integration are needed. It explains that if the anti-derivative is in terms of x, new limits are not required, but if in terms of u, they are necessary. The tutorial provides an example demonstrating both techniques, showing how to calculate integrals with and without changing limits. It concludes by verifying the results using a graphing calculator, emphasizing that both methods yield the same result.

When using the substitution method for definite integrals, under what condition do you need to find new limits of integration?

• When the anti-derivative is in terms of 'u'.

• When the function is linear.

• When the anti-derivative is in terms of 'x'.

• When the integral is indefinite.

In the substitution example, what is the inner function that is substituted with 'u'?

• 4x

• 6x

• 3x^2 + 1

• x^2

What is the differential 'du' equal to in terms of 'dx' in the given example?

• 2x dx

• 3x dx

• 4x dx

• 6x dx

How do you determine the new lower limit of integration in terms of 'u' when x = 0?

• u = 0

• u = 6

• u = 1

• u = 3

What is the new upper limit of integration in terms of 'u' when x = 2?

• u = 9

• u = 15

• u = 13

• u = 11

What is the anti-derivative of 'u^2' in terms of 'u'?

• u^4 / 4

• u^3 / 2

• u^2 / 2

• u^3 / 3

What is the result of evaluating the integral using the limits in terms of 'x'?

• 512

• 1024

• 488

• 256

What is the result of evaluating the integral using the limits in terms of 'u'?

• 512

• 1024

• 488

• 256

Which function is used to verify the integral in terms of 'x' on the graphing calculator?

• 2x(3x^2 + 1)^3

• 6x(3x^2 + 1)

• 4x(3x^2 + 1)^2

• 3x^2 + 1

What is the final result obtained from both methods of integration?

• 512

• 256

• 1024

• 488