Understanding Integration by Substitution

The video tutorial explains how to evaluate an indefinite integral using the method of substitution. It begins by introducing the substitution method, then guides through choosing a substitution variable 'u', differentiating it, and comparing it with the integral. The tutorial continues with substituting 'u' into the integral, simplifying, and integrating with respect to 'u'. Finally, it concludes with the antiderivative result and emphasizes the family of functions derived from the process.

What is the primary reason for using the method of substitution in integration?

• To make the integral more complex

• To fit the integral into a basic formula

• To simplify the integral

• To change the variable of integration

When choosing 'u' in substitution, what should it resemble?

• The entire integrand

• The variable of integration

• A part of the integrand whose derivative is present elsewhere in the integrand

• A constant factor

What rule is applied to find the derivative of 'u' when 'u' is a composite function?

• Chain rule

• Product rule

• Power rule

• Quotient rule

In the given example, what is chosen as 'u'?

• sin(e^3x)

• 5

• e^3x

• 3x

What is the purpose of dividing both sides of the equation by 3 in the substitution process?

• To convert the integral into a definite integral

• To eliminate the factor of 3

• To simplify the differential equation

• To adjust the limits of integration

After substitution, what does the integrand function become?

• e^u

• tan(u)

• sin(u)

• cos(u)

What is the integral of sin(u) with respect to u?

• -sin(u) + C

• sin(u) + C

• -cos(u) + C

• cos(u) + C

What is the final antiderivative in terms of the original variable x?

• -5/3 sin(e^3x) + C

• 5/3 cos(e^3x) + C

• 5/3 sin(e^3x) + C

• -5/3 cos(e^3x) + C

What does the constant 'C' represent in the final antiderivative?

• The derivative of the function

• The constant of integration

• A specific solution

• The variable of integration

What is the family of functions whose derivatives are equal to the given integrand function?

• The derivative

• The indefinite integral

• The definite integral

• The antiderivative