Evaluating Integrals with Substitution

The video tutorial explains how to evaluate a definite integral using integration by substitution. It begins by rewriting the integral with a rational exponent and setting up the substitution by letting u equal the inner function. The integral is then rewritten in terms of u, and the antiderivative is determined. Finally, the definite integral is evaluated by substituting the limits back in terms of x, resulting in a final value of 2/7.

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

• Rewrite the integral with a rational exponent.

• Find the derivative of the function.

• Solve the integral directly.

• Change the limits of integration.

In the substitution method, what is the substitution variable 'u' set equal to?

• The derivative of the function.

• The inner function.

• The outer function.

• The entire integrand.

How do you adjust the differential when using substitution?

• Add the derivative of 'u'.

• Subtract the derivative of 'u'.

• Multiply by the derivative of 'u'.

• Divide by the derivative of 'u'.

What is the integral of 1/u to the 1/2 in terms of 'u'?

• u to the -1/2 divided by -1/2

• u to the 2 divided by 2

• u to the 3/2 divided by 3/2

• u to the 1/2 divided by 1/2

What operation can replace dividing by 1/2 when simplifying the antiderivative?

• Subtracting 2

• Multiplying by 2

• Adding 2

• Dividing by 2

What is the value of 'u' when x equals 2 in the definite integral evaluation?

• 14

• 16

• 18

• 20

What is the square root of 16 in the context of this integral evaluation?

• 5

• 2

• 3

• 4

What is the final result of the definite integral evaluation?

• 4/7

• 3/7

• 2/7

• 1/7

What is the purpose of evaluating the definite integral at the limits 2 and 1?

• To simplify the integrand.

• To calculate the exact area under the curve.

• To determine the constant of integration.

• To find the indefinite integral.