Changing Limits in Definite Integration with Substitution

This 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. The video provides a detailed example, demonstrating both techniques and verifying results using a calculator. The conclusion emphasizes that both methods yield the same result.

When evaluating a definite integral using substitution, do we always need to determine new limits of integration?

• Only if the integral is complex

• It depends on whether we use the anti-derivative in terms of x or u

• No, never

• Yes, always

If we use the anti-derivative in terms of x, do we need to determine new limits of integration?

• No

• Depends on the function

• Only sometimes

• Yes

What is the inner function in the example provided in the video?

• 4x

• 3x^2 + 1

• 6x

• u^2

What is the differential du equal to in the example?

• 3x dx

• 6x dx

• 4x dx

• 2x dx

What should we do if we use the anti-derivative in terms of u?

• Determine new limits of integration for u

• Ignore the limits of integration

• Use the same limits for x and u

• Keep the original limits of integration

What is the new lower limit of integration in terms of u when x is 0?

• 0

• 2

• 1

• 3

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

• 10

• 11

• 12

• 13

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

• u^3 / 3

• u^3 / 2

• u^2 / 2

• u^2 / 3

What is the result of the integral when evaluated using the anti-derivative in terms of x?

• 400

• 450

• 500

• 488

What is the result of the integral when evaluated using the anti-derivative in terms of u?

• 400

• 488

• 450

• 500