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Mastering Definite Integrals: Techniques and Examples

Integration by Parts for Definite Integral with Limits

Have you ever been stuck solving a definite integral with limits? Fear not! Integration by parts may be just what you need to get that solution.

In this article, we’ll explore the rules for solving integration by parts with definite integral limits, how to use limits in the integration by parts formula, and provide some example problems to solidify your understanding.

Rules for Solving Integration by Parts for Definite Integral Limits

Before jumping into the process of solving definite integral limits using integration by parts, it’s essential to know the rules. They are as follows:

1.

Identify which function will be “u” and which will be “dv.” Choose “u” as the function that will become simpler upon differentiation. 2.

Integrate the “v” function to obtain “u.”

3. Differentiate “u” to obtain “du/dx.”

4.

Differentiate “v” to obtain “dv/dx.”

5. Substitute all the values found in steps 2-4 into the integration by parts formula:

(udv) = uv – (vdu)

6.

Apply the limits of integration to the final formula.

Using Limits in Integration by Parts Formula

After applying integration by parts using the steps listed above, the result will be an indefinite integral. To determine the definite integral with limits, we must apply those limits to the formula:

(udv) = uv – (vdu)|a to b

where “a” and “b” are the lower and upper limits of integration, respectively.

Example Problem

Let’s take a look at the following example to better understand the process. Evaluate the definite integrals of x*e^x from 0 to 1.

Solution:

We start by identifying “u” and “dv”:

u = x, dv = e^x

We integrate the “v” function:

v = e^x

We differentiate “u” to obtain “du/dx:”

du/dx = 1

We differentiate “v” to obtain “dv/dx:”

dv/dx = e^x

We substitute all the values found in steps 2-4 into the integration by parts formula:

(udv) = uv – (vdu)

(xe^x) dx= xe^x – (e^x) dx

Next, we integrate the “e^x” function:

(e^x) dx = e^x

Substituting back into the formula, we obtain:

(xe^x) dx= xe^x – e^x + C

Where “C” is the constant of integration. Finally, we apply the limits of integration:

(xe^x) dx|0 to 1 = (1e^1-e^1)-(0e^0-e^0) = e-1

Therefore, the definite integral of x*e^x from 0 to 1 is e-1.

Rule 1: Apply Limits after Indefinite Integration

Now, let’s explore the application of Rule 1, which states that the limits of integration must be applied after we perform the indefinite integration.

Process for Applying Rule 1

1. After performing the indefinite integration, rewrite the integral as follows:

[f(x)] dx|a to b = [F(b) – F(a)]

where “f(x)” is the function being integrated, “F(x)” is the antiderivative of “f(x),” “a” is the lower limit, and “b” is the upper limit.

2. Replace “x” with “b” in the antiderivative of “f(x)” to obtain “F(b),” then do the same for “a” to obtain “F(a).”

3.

Subtract “F(a)” from “F(b)” to obtain the solution.

Example Problem

Let’s apply Rule 1 to the following example:

Evaluate the definite integral of 3x^2 from 0 to 2. Solution:

We first perform the indefinite integration of 3x^2:

(3x^2) dx = x^3

The integral then becomes:

(3x^2) dx|0 to 2 = [2^3 – 0^3] = 8

Therefore, the definite integral of 3x^2 from 0 to 2 is 8.

In conclusion, integrating by parts can be a great tool to solve definite integral limits with ease. The process may seem daunting at first, but by following the rules and guidelines provided in this article and practicing with example problems, you’ll become a pro in no time.

Remember to apply the limits of integration after performing indefinite integration to ensure accuracy. Rule 2: Using Limits with Integrals of Functions

In this section, we will be discussing the use of Rule 2 when working with integrals of functions.

Rule 2 is used when we have an integral with limits that could simplify if one of the limits were different. In this article, we will be breaking down the process for applying Rule 2, and also providing example problems to develop a stronger understanding of the topic.

Process for Applying Rule 2

The process of applying Rule 2 is fairly straightforward. However, It’s important to note that this method is only effective when the integrand is continuous in the interval given.

Let’s say we’re given the integral a to b f(x) dx, and we want to change the upper limit from “b” to “c.” We can represent our integral by dividing the interval [a, c] into two parts: [a, b] and [b, c]. Therefore, we can write the integral as:

a to b f(x) dx + b to c f(x) dx = a to c f(x) dx

The above equation is called the partition formula.

By rearranging this equation, we get:

b to c f(x) dx = a to c f(x) dx – a to b f(x) dx

Hence, to find the integral with the upper limit of “c,” we can subtract the integral with the upper limit of “b” from the integral with the upper limit of “c.”

This process can also be used to change the lower limit of integration from “a” to “c,” as we can represent the integral as follows:

c to b f(x) dx + a to c f(x) dx = a to b f(x) dx

And by rearranging, we can write:

a to c f(x) dx = a to b f(x) dx – c to b f(x) dx

Example Problems

Let’s take a look at some example problems to better understand the application of Rule 2:

Example 1:

Evaluate the definite integral 0 to 2x 2t^3 dt. Solution:

Using the power rule of integration, we have:

0 to 2x 2t^3 dt = [(t^4)/2]|0 to 2x

We can apply Rule 2 by replacing the upper limit “2x” with “t”:

[(t^4)/2]|0 to t – [(t^4)/2]|0 to 0

Simplifying, we obtain:

[(t^4)/2]|0 to t = [(t^4)/2]|0 to 2x = 8x^4

Therefore, the value of the definite integral is 8x^4.

Example 2:

Evaluate the definite integral 1 to 2(3-2x) dx. Solution:

By expanding the integrand, we get:

1 to 2[6x – 2x^2] dx

Applying the power rule of integration to each term, we obtain:

[3x^2]|1 to 2 – [(2x^3)/3]|1 to 2

We can use Rule 2 to change the upper limit from 2 to 4, as follows:

[3x^2]|1 to 4 – [(2x^3)/3]|1 to 4

Substituting, we get:

[3(4^2) – 3(1^2)] – [(2(4^3))/3 – (2(1^3))/3]

Simplifying, we obtain:

48 – (128/3) = 16/3

Therefore, the value of the definite integral is 16/3.

Substitution Integration and Changing Limits

Substitution integration is a technique used to simplify integrals by replacing one function with another. This can be particularly useful when dealing with complex integrals.

When applying substitution integration, the limits of integration also need to be changed accordingly. Let’s take a closer look at the process.

Process for Changing Limits in Substitution Integration

Suppose we have an integral of the form a to b f(g(x))g'(x) dx, and we change the variable within the integral using u = g(x). This can be written as:

f(u) du = F(u) + C

where F(u) is the antiderivative of f(u).

To go back to the original variable x, we can use the substitution u = g(x) and hence the limits on the integral must be modified as well. If a = g(c) and b = g(d), then the integral can be rewritten as:

g(c) to g(d) f(g(x))g'(x) dx = c to d f(u) du

Example Problem

Let’s go through an example to get a stronger understanding of the process:

Example:

Evaluate the definite integral 1 to 2 (x^2 + 4x – 6) dx using substitution integration. Solution:

We start by establishing a substitution expression of u = x + 2:

1 to 2 (x^2 + 4x – 6) dx = u=1+2 u-2)^2 + 4(u-2) – 6 du

Simplifying the integrand, we obtain:

1 to 2 (u^2 – 2u – 2) du

Integrating term by term, we get:

[(u^3)/3 – u^2 – 2u]|1 to 2

Now we can use Rule 2 to modify the limits as follows:

[(2^3)/3 -2^2 -2] – [(1^3)/3 – 1^2 – 2]

Simplifying, we obtain:

(8/3 – 8) – (1/3 – 1) = (-16/3)

Therefore, the value of the definite integral using substitution integration is -16/3.

In conclusion, understanding the different rules and techniques in integration is key in solving complex integrals. By applying Rule 2 with integrals of functions, we can simplify integrals by changing limits.

Similarly, substitution integration is a useful technique for integrating complex functions. By following the process outlined in each method and practicing with example problems, we can improve our understanding and become more skilled in solving integrals.

In this article, we explored various rules and techniques for solving definite integrals, such as integration by parts, using limits with integrals of functions, and substitution integration. We provided a step-by-step process for each method, as well as example problems to help reinforce the concepts.

By understanding and practicing these rules and techniques, one can become more proficient in solving complex integrals. It is important to note that these concepts can be applied in various fields, including physics, economics, and engineering.

Remember to follow the rules carefully, ensure that the integrand is continuous, apply the limits after performing indefinite integration, and change the limits accordingly when performing substitution integration.

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