Understanding Indefinite Integrals and Substitution Method

The video tutorial explains how to evaluate an indefinite integral using the method of substitution. It begins by selecting a part of the integral to substitute, determining the differential, and simplifying the integral. The power rule is applied to integrate with respect to the new variable, and the solution is expressed back in terms of the original variable. The tutorial emphasizes understanding the substitution process and simplifying the integral for easier computation.

What is the first step in using the substitution method for evaluating an indefinite integral?

• Identify the constant in the integral.

• Integrate directly without substitution.

• Differentiate the entire integral.

• Select a part of the integral to be 'u'.

When choosing 'u' for substitution, what should 'u prime' resemble?

• The original function before integration.

• The entire integral function.

• A remaining part of the integral function.

• A constant value.

What is the derivative of the linear function 3x + 7?

• 7

• 3

• 3x + 7

• 3x

After substituting 'u' into the integral, what expression do we get for dx?

• 2 du

• du

• 3 du

• 1/3 du

How is the integral simplified after substitution?

• By rewriting in terms of 'u'.

• By dividing by 3.

• By multiplying by 3.

• By adding a constant.

What is the power of 'u' after applying the power rule of integration?

• 4

• 3

• -3

• -4

What is the coefficient of 'u' after applying the power rule?

• 3/2

• 2/3

• -2/9

• 1/3

How is the anti-derivative expressed using positive exponents?

• 2 divided by 9 times (3x + 7)^3

• Negative 2 divided by 9 times (3x + 7)^-3

• Negative 2 divided by 9 times (3x + 7)^3

• 2 divided by 9 times (3x + 7)^-3

What is the final form of the anti-derivative in terms of 'x'?

• Negative 2/9 times (3x + 7)^-3 + c

• 2/9 times (3x + 7)^-3 + c

• Negative 2/9 times (3x + 7)^3 + c

• 2/9 times (3x + 7)^3 + c

Which of the following is an acceptable form of the anti-derivative?

• Negative 2/9 times (3x + 7)^-3 + c

• 2/9 times (3x + 7)^-3 + c

• 2/9 times (3x + 7)^3 + c

• Negative 2/9 times (3x + 7)^3 + c