31 Lecture

What is the correct substitution for evaluating the definite integral of ?cos(x)sin(x) dx from 0 to ?/2?

a) u = cos(x)

b) u = sin(x)

c) u = cos(x)sin(x)

d) u = ?(1-cos^2(x))

Answer: b) u = sin(x)

Explanation: Using the substitution u = sin(x), we get du/dx = cos(x) dx. Substituting this into the integral and using the limits of integration, we get ?cos(x)sin(x) dx from 0 to ?/2 is equal to ?u du from 0 to 1, which evaluates to 1/2.

What is the correct substitution for evaluating the definite integral of ?x^2sqrt(x+1) dx from 0 to 1?

a) u = x+1

b) u = x^2

c) u = sqrt(x+1)

d) u = x+1/2

Answer: a) u = x+1

Explanation: Using the substitution u = x+1, we get du/dx = 1 dx. Substituting this into the integral and using the limits of integration, we get ?x^2sqrt(x+1) dx from 0 to 1 is equal to ?(u-1)^2 sqrt(u) du from 1 to 2, which can be evaluated using integration by parts.

What is the correct substitution for evaluating the definite integral of ?sec(x)tan(x) dx from 0 to ?/4?

a) u = sec(x)

b) u = tan(x)

c) u = sec(x)tan(x)

d) u = sin(x)/cos(x)

Answer: a) u = sec(x)

Explanation: Using the substitution u = sec(x), we get du/dx = sec(x)tan(x) dx. Substituting this into the integral and using the limits of integration, we get ?sec(x)tan(x) dx from 0 to ?/4 is equal to ?u du from ?2 to 1, which evaluates to ln(?2 + 1).

What is the correct substitution for evaluating the definite integral of ?x^3e^(x^4+1) dx from 0 to 1?

a) u = x^4+1

b) u = e^(x^4+1)

c) u = x^3

d) u = e^x

Answer: a) u = x^4+1

Explanation: Using the substitution u = x^4+1, we get du/dx = 4x^3 dx. Substituting this into the integral and using the limits of integration, we get ?x^3e^(x^4+1) dx from 0 to 1 is equal to (1/4)?e^u du from 1 to 2, which evaluates to (e^2 - e)/4.

What is the correct substitution for evaluating the definite integral of ?(x+1)cos(x^2+2x+1) dx from 0 to 1?

a) u = x^2+2x+1

b) u = cos(x^2+2x+1)

c) u = x+1

d) u = sin(x^2+2x+1)

Answer: a) u = x^2+2x+1

Explanation: Using the substitution u = x^2+2x+1, we

What is the basic principle of substitution in evaluating definite integrals?

Answer: The basic principle of substitution in evaluating definite integrals is to replace the variable of integration with a new variable that is simpler to integrate, then evaluate the integral in terms of the new variable, and finally replace the new variable with the original variable.

What is the purpose of substitution in definite integration?

Answer: The purpose of substitution in definite integration is to simplify the integrand so that it can be more easily integrated.

What are the steps involved in evaluating definite integrals using substitution?

Answer: The steps involved in evaluating definite integrals using substitution are as follows:

Substitute the expression for the new variable in terms of the old variable.

Differentiate the expression for the new variable to find the differential element, and substitute it into the integral.

Simplify the integrand in terms of the new variable.

Evaluate the integral in terms of the new variable.

Substitute the original variable back into the expression to obtain the final answer.

What is the general formula for evaluating definite integrals by substitution?

Answer: The general formula for evaluating definite integrals by substitution is as follows:

?[f(g(x)) * g'(x)] dx = ?f(u) du, where u = g(x).

Can any integral be evaluated using substitution?

Answer: No, not all integrals can be evaluated using substitution. Some integrals require other techniques such as integration by parts or partial fractions.

What is the importance of selecting the appropriate substitution variable?

Answer: Selecting the appropriate substitution variable is important because it simplifies the integrand and makes the integration process easier. Choosing an inappropriate substitution variable can make the integral more difficult or impossible to evaluate.

What are the common substitution formulas used in definite integration?

Answer: The common substitution formulas used in definite integration are trigonometric substitutions, u-substitutions, and exponential substitutions.

What is the difference between indefinite and definite integration using substitution?

Answer: Indefinite integration using substitution involves finding the antiderivative of a function using a substitution technique, while definite integration using substitution involves finding the exact numerical value of a definite integral using a substitution technique.

Can substitution be used to evaluate integrals with more than one variable?

Answer: No, substitution can only be used to evaluate integrals with a single variable.

Can substitution be used to evaluate improper integrals?

Answer: Yes, substitution can be used to evaluate some types of improper integrals. However, it is important to ensure that the limits of integration are appropriate for the given function.

Evaluating Definite Integral by Substitution

Evaluating a definite integral by substitution is a powerful technique that allows us to solve a wide range of integration problems. It is a useful method for evaluating integrals that cannot be evaluated using basic integration rules. This technique involves substituting a new variable into the integral to simplify the expression and make it easier to integrate. The basic idea behind substitution is to transform the integral into a simpler form that can be integrated using known rules. The substitution method can be used when the integrand is a composite function, which is a function that is formed by applying one function to another. For example, if the integrand is of the form f(g(x)), where f and g are functions, then we can use the substitution method to evaluate the integral. The first step in evaluating a definite integral by substitution is to choose an appropriate substitution. This involves identifying a new variable u, which is a function of x, and replacing x with u in the integrand. The choice of substitution should simplify the integrand and make it easier to integrate. The substitution should also allow us to express the limits of integration in terms of the new variable u. Once we have chosen a substitution, we can use the chain rule of differentiation to express the integrand in terms of u. The chain rule states that if f is a composite function of the form f(g(x)), then the derivative of f with respect to x is given by f'(g(x))g'(x). Using this rule, we can express the integrand as a product of a function of u and the derivative of u with respect to x. After expressing the integrand in terms of u, we can replace the limits of integration with the corresponding values of u. This step requires us to convert the limits of integration from their original form in terms of x to their equivalent form in terms of u. This can be done by using the substitution that we made earlier. Once we have the integrand in terms of u and the limits of integration expressed in terms of u, we can evaluate the definite integral by integrating the function of u over the given interval. This step involves applying the basic integration rules to the function of u. After integrating the function of u, we can then substitute back the original variable x to obtain the final result. Let us consider an example to illustrate the use of the substitution method for evaluating a definite integral. Consider the integral ?_0^?/4 sin(x) cos^3(x) dx. To evaluate this integral, we can use the substitution u = cos(x), which implies du/dx = -sin(x) and dx = -du/sin(x). Substituting these expressions into the integrand, we get: ?_0^?/4 sin(x) cos^3(x) dx = ?_1^?2/2 sin(x) cos^2(x) (-du/sin(x)) = -?_1^?2/2 u^2 du The limits of integration have been converted from their original form in terms of x to their equivalent form in terms of u. Now, we can integrate the function of u as follows: -?_1^?2/2 u^2 du = [-u^3/3]_1^?2/2 = -(?2^3/3)/2 + 1/3

Substituting back u = cos(x), we get:

?_0^?/4 sin(x) cos^3(x) dx = -(?2^3/3)/2 + 1/3 = 0.191 In conclusion, the substitution method is a powerful technique for evaluating definite integrals that cannot be evaluated using basic integration rules. This method involves choosing an