Understanding U-Substitution in Integrals

The video tutorial explains how to solve an indefinite integral using u-substitution. It begins by identifying a suitable substitution for u and its derivative. The integral is then rewritten with a scalar factor to facilitate the substitution. The tutorial demonstrates solving the integral and finalizing the solution by unsubstituting back to the original variable.

What is the initial expression we are trying to integrate?

• 7x plus 9

• The square root of 7x plus 9

• The square root of x

• 7x

Why is u-substitution considered for this integral?

• Because it simplifies the integral

• Because it is the only method available

• Because it is a requirement

• Because it is a standard procedure

What is the derivative of u when u is set to 7x plus 9?

• 7x

• 0

• 9

• 7

What mathematical operation is used to introduce the factor of 7 in the integral?

• Addition and subtraction by 7

• Division by 7 only

• Multiplication and division by 7

• Multiplication by 7 only

How is the integral rewritten in terms of u?

• As the integral of the square root of u

• As the integral of u plus 9

• As the integral of u squared

• As the integral of 7u

What is the antiderivative of u to the 1/2 power?

• u to the 1/2 times 2/3

• u to the 3/2 times 3/2

• u to the 1/2 times 3/2

• u to the 3/2 times 2/3

What constant is added to the antiderivative?

• Plus c

• Plus 7

• Plus 1

• Plus 9

What is the final expression of the integral in terms of x?

• 2/21 times (7x plus 9) squared plus c

• 2/21 times (7x plus 9) to the 3/2 power plus c

• 2/21 times (7x plus 9) to the 1/2 power plus c

• 2/21 times (7x plus 9) plus c

What is the purpose of unsubstituting u back to x?

• To express the solution in terms of the original variable

• To introduce a new variable

• To simplify the expression further

• To check the correctness of the solution

What does the final solution demonstrate about u-substitution?

• It can simplify complex integrals

• It is always necessary

• It is a complex process

• It is only for definite integrals