Solving The Integral Of Sin(3x+3) A Step-by-Step Guide

Introduction to Integrating Trigonometric Functions

In the realm of calculus, integrating trigonometric functions is a fundamental skill. Trigonometric functions, such as sine, cosine, tangent, and their reciprocals, appear frequently in mathematical models describing periodic phenomena, oscillations, and wave behavior. Therefore, mastering the integration of these functions is crucial for solving problems in physics, engineering, and various other scientific disciplines. This article delves into the intricacies of integrating ${\sin(3x+3)}$, providing a step-by-step guide to tackle this specific integral and similar problems. Before diving into the specifics, it's essential to grasp the basic integration rules for trigonometric functions. The integral of ${\sin(x)}$ is ${-\cos(x) + C}$, where ${C}$ is the constant of integration. This constant is added because the derivative of a constant is zero, meaning that when we reverse the process of differentiation (i.e., integration), we need to account for any possible constant term. Understanding this basic rule is the cornerstone for tackling more complex trigonometric integrals. Furthermore, it's important to recognize the chain rule in reverse, which often comes into play when integrating composite functions, such as ${\sin(3x+3)}$. The chain rule states that the derivative of a composite function ${f(g(x))}$ is ${f'(g(x)) imes g'(x)}$. In integration, we need to identify the composite function and adjust for the derivative of the inner function. This adjustment typically involves dividing by the derivative of the inner function, as we will see in the following sections. The process of integrating ${\sin(3x+3)}$ involves a technique known as u-substitution, which simplifies the integral by making it easier to recognize and apply the basic integration rules. This method is particularly useful when dealing with trigonometric functions with linear arguments, such as ${3x+3}$. By substituting a new variable for the linear argument, we transform the integral into a more manageable form. The key to successful integration lies in practice and a solid understanding of the underlying principles. By working through various examples and exercises, one can develop the intuition and skills necessary to tackle a wide range of integration problems. This article aims to provide that foundation, offering a detailed explanation of the integration of ${\sin(3x+3)}$ and related concepts.

Breaking Down the Integral: A Step-by-Step Approach

To effectively integrate ${\sin(3x+3)}$, we employ the method of u-substitution, a powerful technique for simplifying integrals. U-substitution is essentially the reverse of the chain rule in differentiation. It allows us to transform a complex integral into a simpler one by substituting a part of the integrand with a new variable, ${u}$. In this case, the argument of the sine function, ${3x+3}$, is the ideal candidate for substitution. Let's set ${u = 3x + 3}$. The next step is to find the differential of ${u}$, denoted as ${du}$. We differentiate both sides of the equation ${u = 3x + 3}$ with respect to ${x}$: $\fracdu}{dx} = 3$ This gives us ${du = 3 dx}$. However, our original integral contains ${dx}$, not ${3 dx}$. Therefore, we need to isolate ${dx}$ by dividing both sides of the equation by 3 $\frac{13} du = dx$ Now we have all the necessary components to perform the substitution. We replace ${3x + 3}$ with ${u}$ and ${dx}$ with ${\frac{1}{3} du}$ in the original integral $\int \sin(3x+3) dx = \int \sin(u) \cdot \frac{13} du$ We can pull the constant ${\frac{1}{3}}$ out of the integral $\frac{13} \int \sin(u) du$ This integral is now in a standard form that we can easily solve. The integral of ${\sin(u)}$ with respect to ${u}$ is ${-\cos(u) + C}$, where ${C}$ is the constant of integration $\frac{13} \int \sin(u) du = \frac{1}{3} (-\cos(u) + C_1)$ Here, we use ${C_1}$ to represent the constant of integration to avoid confusion. We can distribute the ${\frac{1}{3}}$ $-\frac{13} \cos(u) + \frac{1}{3}C_1$ Since ${\frac{1}{3}C_1}$ is still an arbitrary constant, we can replace it with a single constant ${C}$ $-\frac{13} \cos(u) + C$ The final step is to substitute back the original variable ${x}$. We replace ${u}$ with ${3x + 3}$ $-\frac{1{3} \cos(3x+3) + C$ Thus, the integral of ${\sin(3x+3)}$ is ${-\frac{1}{3} \cos(3x+3) + C}$.

U-Substitution: The Key to Simplifying Integrals

The u-substitution method is a pivotal technique in integral calculus, designed to simplify integrals by replacing a complex expression with a single variable, ${u}$. This method is particularly effective when dealing with composite functions, where one function is nested inside another. The essence of u-substitution lies in the reverse application of the chain rule, allowing us to unravel the composition and make the integral more manageable. To illustrate the power of u-substitution, let's revisit the integral ${\int \sin(3x+3) dx}$. The argument of the sine function, ${3x+3}$, is a linear function that complicates the integration process. By substituting ${u = 3x + 3}$, we transform the integral into a simpler form involving ${\sin(u)}$. The process involves several key steps. First, we identify the appropriate substitution. In this case, ${u = 3x + 3}$ is a natural choice because it simplifies the sine function's argument. Next, we find the differential ${du}$ by differentiating ${u}$ with respect to ${x}$: $\fracdu}{dx} = 3$ This gives us ${du = 3 dx}$. Crucially, we need to express ${dx}$ in terms of ${du}$, so we solve for ${dx}$ $dx = \frac{13} du$ Now we substitute ${u}$ and ${dx}$ into the original integral $\int \sin(3x+3) dx = \int \sin(u) \cdot \frac{13} du$ The constant ${\frac{1}{3}}$ can be pulled out of the integral $\frac{13} \int \sin(u) du$ This resulting integral is much simpler to evaluate. The integral of ${\sin(u)}$ is ${-\cos(u) + C}$, where ${C}$ is the constant of integration $\frac{13} \int \sin(u) du = -\frac{1}{3} \cos(u) + C$ Finally, we substitute back the original variable ${x}$ $-\frac{1{3} \cos(u) + C = -\frac{1}{3} \cos(3x+3) + C$ Thus, the integral of ${\sin(3x+3)}$ is ${-\frac{1}{3} \cos(3x+3) + C}$. U-substitution is not limited to trigonometric functions; it can be applied to a wide range of integrals. The key is to identify a suitable substitution that simplifies the integrand. This often involves recognizing composite functions or expressions that appear repeatedly in the integral. By mastering u-substitution, one can significantly expand their ability to solve complex integrals. The technique is a cornerstone of integral calculus and a valuable tool for any mathematician or scientist.

The Importance of the Constant of Integration

In the realm of integral calculus, the constant of integration, denoted as ${C}$, plays a critical role. It arises from the fundamental nature of indefinite integrals, which represent a family of functions that differ only by a constant. To understand why the constant of integration is essential, we must consider the reverse relationship between differentiation and integration. Differentiation is a process that finds the rate of change of a function, while integration is the reverse process, finding a function given its rate of change. However, the process of differentiation eliminates constant terms. For example, the derivative of ${x^2 + 5}$ is ${2x}$, and the derivative of ${x^2 - 3}$ is also ${2x}$. When we integrate ${2x}$, we obtain ${x^2}$, but we lose information about the original constant term. Therefore, we must add a constant of integration, ${C}$, to represent all possible constant terms that could have been present in the original function. The indefinite integral of a function ${f(x)}$ is written as: $\int f(x) dx = F(x) + C$ where ${F(x)}$ is an antiderivative of ${f(x)}$, meaning that ${F'(x) = f(x)}$. The constant ${C}$ represents an arbitrary constant, which can take any real value. This means that the indefinite integral represents an infinite family of functions, each differing by a constant vertical shift. In the context of our integral, ${\int \sin(3x+3) dx}$, we found the antiderivative to be ${-\frac{1}{3} \cos(3x+3)}$. However, the complete solution includes the constant of integration: $-\frac{1}{3} \cos(3x+3) + C$ The constant ${C}$ is crucial because it accounts for all possible vertical shifts of the function. Without it, we would only have one specific antiderivative, rather than the entire family of antiderivatives. The constant of integration has significant implications in various applications of calculus. For example, in physics, when solving differential equations that describe the motion of an object, the constant of integration represents the initial conditions of the system, such as the initial position or velocity. In engineering, it can represent a baseline value or a reference point. In summary, the constant of integration is not just a mathematical formality; it is an essential component of indefinite integrals that represents the inherent ambiguity in the reverse process of differentiation. It ensures that we capture the entire family of antiderivatives and allows us to apply integration to solve real-world problems accurately.

Common Mistakes to Avoid When Integrating Trigonometric Functions

Integrating trigonometric functions can be challenging, and it's easy to make mistakes if one is not careful. One of the common mistakes is forgetting the constant of integration, ${C}$. As discussed earlier, the constant of integration is crucial because it represents the family of antiderivatives. Omitting it results in an incomplete solution. Another frequent error is incorrectly applying the chain rule in reverse. When using u-substitution, it's essential to correctly find the differential ${du}$ and substitute it into the integral. For instance, in the integral ${\int \sin(3x+3) dx}$, if we forget to divide by the derivative of the inner function (which is 3), we would obtain the wrong result. The correct substitution is ${u = 3x + 3}$, ${du = 3 dx}$, and ${dx = \frac{1}{3} du}$. If we mistakenly write ${dx = du}$, we would end up with an incorrect integral. Another mistake is misapplying the basic integration formulas for trigonometric functions. It's crucial to remember that the integral of ${\sin(x)}$ is ${-\cos(x) + C}$, and the integral of ${\cos(x)}$ is ${\sin(x) + C}$. Confusing these formulas can lead to incorrect answers. For example, incorrectly stating that the integral of ${\sin(x)}$ is ${\cos(x) + C}$ would be a significant error. Additionally, students often struggle with the signs when integrating trigonometric functions. The negative sign in the integral of sine can be particularly tricky. It's helpful to remember the derivatives of sine and cosine to avoid sign errors. Since the derivative of ${-\cos(x)}$ is ${\sin(x)}$, the integral of ${\sin(x)}$ must be ${-\cos(x) + C}$. A further mistake is failing to simplify the integral after performing u-substitution. After substituting back the original variable, it's good practice to check if the result can be simplified further. In some cases, trigonometric identities can be used to simplify the expression. For instance, if we had an expression involving ${\cos(2x)}$, we might use the double-angle formula to simplify it. Finally, neglecting to check the answer by differentiating is a common oversight. Differentiation is the reverse process of integration, so differentiating the result should yield the original integrand. If it doesn't, there is an error in the integration process. By being mindful of these common mistakes and practicing regularly, one can improve their skills in integrating trigonometric functions and avoid these pitfalls.

Conclusion: Mastering Trigonometric Integration

In conclusion, mastering trigonometric integration is a crucial aspect of calculus, essential for solving a wide range of mathematical and scientific problems. This article has provided a detailed guide to integrating ${\sin(3x+3)}$, demonstrating the power and versatility of the u-substitution method. By breaking down the integral into manageable steps, we have shown how to transform a seemingly complex problem into a straightforward application of basic integration rules. We began by introducing the fundamental concepts of trigonometric integration, emphasizing the importance of understanding the basic integration rules for sine, cosine, and other trigonometric functions. We then delved into the step-by-step process of integrating ${\sin(3x+3)}$, highlighting the role of u-substitution in simplifying the integral. The substitution ${u = 3x + 3}$ allowed us to transform the integral into a standard form, which we could easily solve. We also discussed the significance of the constant of integration, ${C}$, and its role in representing the family of antiderivatives. The constant of integration is not merely a formality; it is a critical component that accounts for all possible vertical shifts of the function. Furthermore, we addressed common mistakes to avoid when integrating trigonometric functions, such as forgetting the constant of integration, misapplying the chain rule, and making sign errors. By being aware of these pitfalls and practicing diligently, one can improve their accuracy and proficiency in integration. The u-substitution method is a powerful tool that extends beyond trigonometric functions. It can be applied to a wide variety of integrals, making it a cornerstone of integral calculus. The key to successful integration lies in practice and a solid understanding of the underlying principles. By working through numerous examples and exercises, one can develop the intuition and skills necessary to tackle complex integration problems. In summary, integrating ${\sin(3x+3)}$ serves as a valuable exercise in mastering trigonometric integration techniques. By understanding the principles of u-substitution, the importance of the constant of integration, and common pitfalls to avoid, students can build a strong foundation in calculus and apply these skills to solve real-world problems in various fields. This article has aimed to provide a comprehensive guide, equipping readers with the knowledge and confidence to tackle trigonometric integrals effectively.