Evaluating The Integral Of (2x + 3)^10 A Step-by-Step Guide

In this comprehensive guide, we will delve into the evaluation of the integral $\int(2x + 3)^{10} dx$. This problem falls under the category of mathematics, specifically integral calculus, and involves finding the antiderivative of a composite function raised to a power. Mastering such integrals is crucial for students and professionals in various fields, including physics, engineering, and economics, where calculus serves as a fundamental tool. This article aims to provide a step-by-step solution, ensuring clarity and understanding for readers of all backgrounds. We will explore the underlying concepts, apply the appropriate techniques, and present the final result in a clear and concise manner. Our focus will be on the method of substitution, a powerful tool for simplifying integrals of this form. We will break down the process into manageable steps, making it easy to follow along and grasp the core principles involved. By the end of this guide, you will not only be able to solve this specific integral but also gain a deeper understanding of integration techniques applicable to a wide range of problems. So, let's embark on this mathematical journey and unravel the solution together. The process of integration can sometimes seem daunting, but with a systematic approach and a solid grasp of the fundamental rules, it becomes a much more manageable task. Remember, practice is key to mastering any mathematical concept, and this guide will provide you with the necessary tools and knowledge to tackle similar problems with confidence. So, let's dive in and begin our exploration of this fascinating integral.

Understanding the Integral

To effectively evaluate the integral $\int(2x + 3)^{10} dx$, it's essential to first understand the structure of the integrand and identify the appropriate integration technique. The integrand, $(2x + 3)^{10}$, is a composite function where a linear expression $(2x + 3)$ is raised to the power of 10. This form suggests that the method of substitution, also known as u-substitution, will be particularly effective. U-substitution is a technique used to simplify integrals by replacing a part of the integrand with a new variable, often denoted as u. This substitution aims to transform the integral into a simpler form that can be easily integrated using standard integration rules. The key to successful u-substitution lies in choosing the right expression to substitute for u. In this case, the inner function $(2x + 3)$ is an excellent candidate for substitution. By setting u equal to $(2x + 3)$, we can simplify the integrand and make it easier to find the antiderivative. The power of 10 outside the parenthesis further confirms that the power rule of integration will play a crucial role once we've simplified the integral through substitution. The power rule states that the integral of x raised to the power of n is x raised to the power of (n + 1) divided by (n + 1), plus a constant of integration. However, we need to account for the chain rule in reverse due to the composite nature of the function. This is where the substitution method shines, allowing us to unwind the chain rule and find the correct antiderivative. Before we proceed with the substitution, it's worth noting that a direct application of the power rule without considering the inner function would lead to an incorrect result. This highlights the importance of recognizing composite functions and employing appropriate techniques like u-substitution. The integral represents the area under the curve of the function $(2x + 3)^{10}$, and finding the antiderivative allows us to calculate this area over a given interval. This has numerous applications in various fields, including calculating the displacement of an object given its velocity function or determining the accumulated change in a quantity over time. Understanding the context and applications of integration can further motivate the learning process and highlight the practical significance of these mathematical tools.

Step-by-Step Solution Using U-Substitution

Now, let's proceed with the step-by-step solution of the integral $\int(2x + 3)^{10} dx$ using the u-substitution method. This technique is particularly effective for integrals involving composite functions, where one function is nested inside another. In our case, the function $(2x + 3)$ is nested within the power function, making u-substitution the ideal approach.

1. Choose a Suitable Substitution

The first crucial step is to identify the part of the integrand that, when substituted, will simplify the integral. As discussed earlier, the inner function $(2x + 3)$ is a prime candidate. We set:

$$u = 2x + 3$$

This substitution replaces the more complex expression $(2x + 3)$ with a single variable, u, making the integrand more manageable. The choice of u is often guided by the chain rule in reverse. We look for a function whose derivative is also present in the integral, or at least a constant multiple of it.

2. Find the Differential du

Next, we need to find the differential du, which represents the derivative of u with respect to x, multiplied by dx. Differentiating both sides of the equation $u = 2x + 3$ with respect to x, we get:

$$\frac{du}{dx} = 2$$

Multiplying both sides by dx, we obtain:

$$du = 2 dx$$

This differential du will be used to replace dx in the original integral. Notice that we have a factor of 2 in du, while the original integral has just dx. We need to adjust for this constant factor.

3. Adjust for Constants

To replace dx in the integral, we need to isolate it in the equation $du = 2 dx$. Dividing both sides by 2, we get:

$$\frac{1}{2} du = dx$$

Now we have an expression for dx in terms of du, which we can substitute into the integral.

4. Substitute and Rewrite the Integral

We can now substitute u for $(2x + 3)$ and $\frac{1}{2} du$ for dx in the original integral. This gives us:

$$\int(2x + 3)^{10} dx = \int u^{10} \cdot \frac{1}{2} du$$

We can pull the constant factor $\frac{1}{2}$ outside the integral sign:

$$= \frac{1}{2} \int u^{10} du$$

The integral is now in a much simpler form, involving only a power of u. This is the key advantage of using u-substitution.

5. Integrate with Respect to u

Now we can apply the power rule of integration to find the antiderivative of $u^{10}$. The power rule states that $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where C is the constant of integration. Applying this rule to our integral, we get:

$$\frac{1}{2} \int u^{10} du = \frac{1}{2} \cdot \frac{u^{11}}{11} + C$$

Simplifying, we have:

$$= \frac{u^{11}}{22} + C$$

This is the antiderivative in terms of u. However, we need to express the final result in terms of the original variable, x.

6. Substitute Back for x

Finally, we substitute back the original expression for u, which was $u = 2x + 3$. This gives us the antiderivative in terms of x:

$$\frac{u^{11}}{22} + C = \frac{(2x + 3)^{11}}{22} + C$$

This is the final result of the integration. We have successfully found the antiderivative of $(2x + 3)^{10}$ using the u-substitution method.

Final Answer

Therefore, the integral of $\int(2x + 3)^{10} dx$ is:

$$\frac{(2x + 3)^{11}}{22} + C$$

Where C represents the constant of integration. This result represents the family of functions whose derivative is $(2x + 3)^{10}$. The constant of integration, C, accounts for the fact that the derivative of a constant is always zero, so there are infinitely many antiderivatives that differ only by a constant term. The solution highlights the power and elegance of the u-substitution method in simplifying complex integrals. By carefully choosing the substitution and following the steps outlined above, we can successfully evaluate a wide range of integrals involving composite functions. The process of u-substitution effectively reverses the chain rule of differentiation, allowing us to find the antiderivative of a function that was obtained by applying the chain rule. Understanding this connection between differentiation and integration is crucial for mastering calculus. The final answer can be verified by differentiating it with respect to x. The derivative of $\frac{(2x + 3)^{11}}{22} + C$ should be equal to the original integrand, $(2x + 3)^{10}$. This verification step provides a valuable check on the correctness of the solution.

Conclusion

In conclusion, we have successfully evaluated the integral $\int(2x + 3)^{10} dx$ using the u-substitution method. This technique proves to be a powerful tool for simplifying integrals of composite functions. By carefully selecting the substitution u = 2x + 3, finding the differential du, and adjusting for constants, we transformed the integral into a simpler form that could be easily solved using the power rule of integration. The final result, $\frac{(2x + 3)^{11}}{22} + C$, represents the antiderivative of the given function, where C is the constant of integration. This constant emphasizes the family of functions that satisfy the original integral, highlighting a fundamental aspect of indefinite integration. The process of u-substitution not only provides a solution to this specific problem but also offers a valuable framework for tackling a wide variety of integrals involving composite functions. The key to mastering this technique lies in recognizing the inner function within the integrand and choosing an appropriate substitution that simplifies the expression. Practice is essential for developing proficiency in u-substitution and other integration methods. By working through various examples and understanding the underlying principles, you can build confidence and expertise in solving complex integrals. The ability to evaluate integrals is a fundamental skill in calculus and has numerous applications in various fields, including physics, engineering, economics, and statistics. From calculating areas and volumes to modeling physical phenomena and analyzing data, integration plays a crucial role in solving real-world problems. Therefore, a solid understanding of integration techniques is essential for anyone pursuing a career in these areas. This guide has provided a comprehensive and step-by-step approach to evaluating the integral of (2x + 3)^10, equipping you with the knowledge and skills to tackle similar problems with confidence. Remember to always double-check your work and verify your results by differentiating the antiderivative to ensure it matches the original integrand. With consistent effort and practice, you can master the art of integration and unlock its vast potential for solving complex problems in mathematics and beyond.