Definite Integrals A Step By Step Solution Guide

Definite integrals are a fundamental concept in calculus, providing a way to calculate the area under a curve between two specified limits. In this comprehensive guide, we will explore the evaluation of several definite integrals, providing step-by-step solutions and explanations. Understanding definite integrals is crucial for various applications in mathematics, physics, engineering, and other fields. This article aims to provide a clear and detailed explanation of how to solve definite integrals, ensuring a solid understanding for both students and professionals. The examples covered here will help illustrate the techniques and principles involved in evaluating these integrals. We will delve into the methods required to solve each integral, ensuring a thorough grasp of the subject matter. By the end of this guide, you will be equipped with the knowledge and skills to tackle a wide range of definite integral problems. Let's begin our journey into the world of definite integrals and uncover the solutions to the problems presented.

Problem 26: Evaluating

$$\int_{0}^{2} (3x^2 - 3x + 7)dx$$

To solve this definite integral, we will apply the power rule for integration and the fundamental theorem of calculus. This problem is a classic example of how to integrate a polynomial function over a given interval. The key steps involve finding the antiderivative of the polynomial and then evaluating it at the upper and lower limits of integration. This process demonstrates the core principles of integral calculus and its application to finding the area under a curve. Understanding this process is fundamental for more advanced topics in calculus and its applications in various fields. Let's break down the solution step by step to ensure clarity and comprehension.

Step 1: Find the Antiderivative

First, we find the antiderivative of the integrand ${3x^2 - 3x + 7}$. We integrate each term separately:

$$\int 3x^2 dx = x^3$$

$$\int -3x dx = -\frac{3}{2}x^2$$

$$\int 7 dx = 7x$$

Combining these, the antiderivative is:

$$F(x) = x^3 - \frac{3}{2}x^2 + 7x$$

This step is crucial as it sets the foundation for applying the fundamental theorem of calculus. The antiderivative represents the function whose derivative is the integrand. Finding the antiderivative correctly is essential for obtaining the correct result. The power rule of integration is applied here, which is a fundamental technique in calculus. The antiderivative is the core of evaluating definite integrals, and mastering this step is key to success in integral calculus.

Step 2: Apply the Fundamental Theorem of Calculus

The fundamental theorem of calculus states that:

$$\int_{a}^{b} f(x) dx = F(b) - F(a)$$

where ${F(x)}$ is the antiderivative of ${f(x)}$. In our case, ${a = 0}$, ${b = 2}$, and ${F(x) = x^3 - \frac{3}{2}x^2 + 7x}$. We evaluate ${F(2)}$ and ${F(0)}$:

$$F(2) = (2)^3 - \frac{3}{2}(2)^2 + 7(2) = 8 - 6 + 14 = 16$$

$$F(0) = (0)^3 - \frac{3}{2}(0)^2 + 7(0) = 0$$

So, the definite integral is:

$$\int_{0}^{2} (3x^2 - 3x + 7)dx = F(2) - F(0) = 16 - 0 = 16$$

This step completes the evaluation of the definite integral. The fundamental theorem of calculus provides the direct link between differentiation and integration. By evaluating the antiderivative at the limits of integration and subtracting, we find the exact value of the definite integral. This result represents the area under the curve of the function ${3x^2 - 3x + 7}$ between ${x = 0}$ and ${x = 2}$. The accuracy of this step is crucial, and careful calculation ensures the correct final answer. This process highlights the power and elegance of calculus in solving real-world problems.

Problem 27: Evaluating

$$\int_{4}^{10} \frac{dx}{x+2}$$

This definite integral involves a rational function. To solve it, we will use a u-substitution and the properties of logarithms. This problem showcases the importance of recognizing suitable substitutions to simplify integrals. The presence of a rational function often suggests the use of logarithmic integration. This type of integral is commonly encountered in various applications, such as calculating areas and solving differential equations. Understanding the technique of u-substitution is essential for mastering integral calculus. Let’s break down the solution step by step.

Step 1: Perform u-substitution

Let ${u = x + 2}$, then ${du = dx}$. When ${x = 4}$, ${u = 4 + 2 = 6}$, and when ${x = 10}$, ${u = 10 + 2 = 12}$. The integral becomes:

$$\int_{6}^{12} \frac{du}{u}$$

The u-substitution transforms the integral into a simpler form that is easier to evaluate. This step is a critical technique in integral calculus, allowing us to handle complex integrals by changing the variable of integration. The ability to identify appropriate substitutions is a key skill in solving a wide range of integral problems. This substitution simplifies the integrand to a basic form that can be directly integrated using known rules. The change of limits is also essential to ensure the integral is evaluated correctly in terms of the new variable.

Step 2: Integrate and Evaluate

The integral of ${\frac{1}{u}}$ is ${\ln |u|}$. So, we have:

$$\int_{6}^{12} \frac{du}{u} = \ln |u| \Big|_{6}^{12} = \ln(12) - \ln(6)$$

Using the logarithm property ${\ln(a) - \ln(b) = \ln(\frac{a}{b})}$, we get:

$$\ln(12) - \ln(6) = \ln(\frac{12}{6}) = \ln(2)$$

Therefore, the definite integral is:

$$\int_{4}^{10} \frac{dx}{x+2} = \ln(2)$$

This step completes the evaluation of the integral using the properties of logarithms. The natural logarithm is a fundamental function in calculus, and its properties are crucial for simplifying expressions and obtaining final answers. The application of the logarithmic property in this step is a common technique used to simplify the result. The final answer, ${\ln(2)}$, represents the exact value of the definite integral. This problem highlights the importance of understanding logarithmic functions and their properties in the context of integral calculus. The precise evaluation in this step ensures an accurate result, showcasing the elegance of calculus in solving mathematical problems.

Final Answer:

The correct answer is D. ${\ln 2}$.

Problem 28: Evaluating

$$\int_{1}^{8} \sqrt{1+3x}dx$$

This definite integral involves a square root function. To solve it, we will again use u-substitution. This problem further reinforces the technique of u-substitution and its applicability to integrals involving radical functions. Such integrals are frequently encountered in physics and engineering applications, such as calculating arc lengths and surface areas. Recognizing the appropriate substitution is crucial for simplifying the integral and making it solvable. The step-by-step solution will demonstrate how to handle this type of integral efficiently.

Step 1: Perform u-substitution

Let ${u = 1 + 3x}$, then ${du = 3 dx}$, and ${dx = \frac{1}{3} du}$. When ${x = 1}$, ${u = 1 + 3(1) = 4}$, and when ${x = 8}$, ${u = 1 + 3(8) = 25}$. The integral becomes:

$$\int_{4}^{25} \sqrt{u} \cdot \frac{1}{3} du = \frac{1}{3} \int_{4}^{25} u^{\frac{1}{2}} du$$

The u-substitution simplifies the integral by transforming the expression under the square root. This step is a pivotal technique in integral calculus, particularly when dealing with composite functions. The correct choice of ${u}$ allows for the integral to be expressed in a more manageable form. The adjustment of the differential ${dx}$ to ${\frac{1}{3} du}$ is essential for maintaining the equality of the integral. The change in limits of integration from ${x}$ to ${u}$ is crucial for the correct evaluation of the definite integral. This step highlights the power of substitution in simplifying complex integrals.

Step 2: Integrate and Evaluate

Now, we integrate ${u^{\frac{1}{2}}}$ with respect to ${u}$:

$$\frac{1}{3} \int_{4}^{25} u^{\frac{1}{2}} du = \frac{1}{3} \cdot \frac{2}{3} u^{\frac{3}{2}} \Big|_{4}^{25} = \frac{2}{9} u^{\frac{3}{2}} \Big|_{4}^{25}$$

Evaluate at the limits:

$$\frac{2}{9} (25^{\frac{3}{2}} - 4^{\frac{3}{2}}) = \frac{2}{9} (125 - 8) = \frac{2}{9} (117) = 26$$

Therefore, the definite integral is:

$$\int_{1}^{8} \sqrt{1+3x}dx = 26$$

This step completes the evaluation of the definite integral. The power rule for integration is applied to find the antiderivative of ${u^{\frac{1}{2}}}$. Evaluating the antiderivative at the upper and lower limits of integration and subtracting gives the final result. This process demonstrates the fundamental theorem of calculus in action. The result, 26, represents the exact value of the area under the curve of the function ${\sqrt{1+3x}}$ between ${x = 1}$ and ${x = 8}$. The accurate application of the power rule and correct evaluation at the limits ensure the precise answer. This problem showcases the importance of careful computation in integral calculus.

Final Answer:

The correct answer is C. 26.

Problem 29: Evaluating

$$\int_{1}^{2} e^x dx$$

This definite integral involves the exponential function. To solve it, we will use the fact that the integral of ${e^x}$ is simply ${e^x}$. This problem highlights the simplicity and elegance of integrating exponential functions. The exponential function is fundamental in calculus and appears in numerous applications, including growth and decay models, probability, and physics. Understanding how to integrate ${e^x}$ is essential for a strong foundation in calculus. Let’s go through the step-by-step solution.

Step 1: Find the Antiderivative

The antiderivative of ${e^x}$ is ${e^x}$. This is a fundamental rule in calculus, making the integration straightforward.

This step is crucial because it establishes the function that will be evaluated at the limits of integration. The exponential function ${e^x}$ is unique in that its derivative and antiderivative are the same. This property simplifies the integration process significantly. The antiderivative forms the basis for applying the fundamental theorem of calculus. A clear understanding of this step is essential for solving integrals involving exponential functions. The simplicity of this step underscores the elegance of calculus in handling exponential integrals.

Step 2: Apply the Fundamental Theorem of Calculus

We apply the fundamental theorem of calculus:

$$\int_{1}^{2} e^x dx = e^x \Big|_{1}^{2} = e^2 - e^1$$

So, the definite integral is:

$$\int_{1}^{2} e^x dx = e^2 - e$$

This step completes the evaluation of the definite integral. The fundamental theorem of calculus provides the direct method for computing definite integrals. By evaluating the antiderivative at the limits of integration and subtracting, we obtain the exact value of the integral. This result represents the area under the curve of the exponential function ${e^x}$ between ${x = 1}$ and ${x = 2}$. The clarity and precision of this step demonstrate the power of calculus in solving mathematical problems. The final answer, ${e^2 - e}$, is a concise and exact representation of the integral’s value. This problem exemplifies the ease with which exponential functions can be integrated using basic calculus principles.

Final Answer:

The correct answer is A. ${e^2 - e}$.

In this comprehensive guide, we have explored the evaluation of several definite integrals, each requiring a specific technique. From integrating polynomials to rational and radical functions, and finally, exponential functions, we have covered a range of common integral types. Mastering these techniques is crucial for success in calculus and its applications. Definite integrals are a fundamental tool in various fields, and a solid understanding of their evaluation is essential for both academic and professional pursuits. By working through these examples, you should now have a stronger grasp of how to approach and solve definite integrals. Remember, practice is key to mastering these concepts, so continue to work through problems and solidify your understanding. This guide serves as a valuable resource for anyone looking to deepen their knowledge of integral calculus and its applications.