Integrating 3x^2 - 4x + Sin(x) A Step-by-Step Solution

Integrating polynomial and trigonometric functions is a fundamental skill in calculus. This article delves into the step-by-step process of integrating the expression $3x^2 - 4x + \sin x$, offering a comprehensive guide suitable for students and anyone interested in deepening their understanding of integral calculus. We'll break down each term, apply relevant integration rules, and provide explanations to ensure clarity. This integration problem combines polynomial terms ($3x^2$ and $-4x$) with a trigonometric term ($\sin x$), requiring us to use different integration rules for each. The integral of a sum (or difference) of functions is the sum (or difference) of their integrals, allowing us to tackle each term separately. This article will explore the integral of $3x^2 - 4x + \sin x$ by applying the power rule for integration and the integral of the sine function. The power rule states that the integral of $x^n$ is $(x^{n+1})/(n+1)$, provided that $n$ is not equal to -1. We'll apply this to the polynomial terms. The integral of $\sin x$ is $-\cos x$, a standard result derived from the definition of integration as the reverse process of differentiation. Throughout this explanation, we will emphasize the importance of the constant of integration, denoted by $C$. This constant arises because the derivative of a constant is zero, meaning that when we find an antiderivative, any constant could be added to it without changing its derivative. Therefore, including $C$ is crucial for expressing the most general form of the integral. Let's embark on this journey of integration with a clear step-by-step methodology, ensuring a solid grasp of the underlying principles and techniques.

Step-by-Step Integration Process

To effectively integrate the expression $3x^2 - 4x + \sin x$, we will dissect the problem into manageable parts, applying the appropriate integration rules to each term individually. This approach leverages the property that the integral of a sum (or difference) is the sum (or difference) of the integrals. This property allows us to handle each term—$3x^2$, $-4x$, and $\sin x$—separately and then combine the results. This makes the integration process more organized and less prone to errors. We'll start by focusing on the first term, $3x^2$. This is a polynomial term, and we can integrate it using the power rule. The power rule states that the integral of $x^n$ with respect to $x$ is $(x^{n+1})/(n+1)$, where $n$ is any real number except -1. The constant multiple rule also comes into play here, which says that the integral of a constant times a function is the constant times the integral of the function. Next, we'll address the second term, $-4x$. This term is also a polynomial, and we will again use the power rule for integration. Note that $-4x$ can be thought of as $-4x^1$, which allows us to apply the power rule in a straightforward manner. The constant multiple rule will be used again here. Lastly, we will deal with the trigonometric term, $\sin x$. The integral of $\sin x$ with respect to $x$ is a standard result in calculus and is equal to $-\cos x$. This result can be derived from the definition of integration as the reverse process of differentiation. Once we have integrated each term individually, we will combine the results. It's crucial to remember to add the constant of integration, $C$, at the end. This constant accounts for the fact that the derivative of a constant is zero, meaning that any constant could be added to the antiderivative without changing the derivative. By following this step-by-step process, we can systematically find the integral of $3x^2 - 4x + \sin x$ and ensure a comprehensive understanding of the method.

Integrating $3x^2$

Integrating the term $3x^2$ is a quintessential application of the power rule in calculus. The power rule for integration is a fundamental concept that states the integral of $x^n$ with respect to $x$ is given by $(x^{n+1})/(n+1)$, where $n$ is any real number except -1. This rule is widely used for integrating polynomial functions, and it is a cornerstone of integral calculus. In our case, we have $3x^2$, which is a polynomial term where the exponent of $x$ is 2. To integrate this term, we first recognize the constant multiple, which is 3. The constant multiple rule in integration allows us to pull this constant out of the integral. This simplifies the problem by allowing us to focus on integrating $x^2$ first and then multiplying the result by 3. Applying the power rule to $x^2$, we increase the exponent by 1, making it 3, and then divide by the new exponent, which is also 3. Therefore, the integral of $x^2$ is $(x^3)/3$. Now, we multiply this result by the constant multiple 3 that we initially factored out. This gives us $3 * (x^3)/3$, which simplifies to $x^3$. It's crucial to remember that integration is the reverse process of differentiation. To verify our result, we can differentiate $x^3$ with respect to $x$. Using the power rule for differentiation, we get $3x^2$, which is the original term we started with. This confirms that our integration was performed correctly. This step-by-step breakdown of integrating $3x^2$ showcases the power rule in action and provides a clear example of how to apply this rule to polynomial terms. This method not only gives us the correct antiderivative but also reinforces the fundamental principles of integral calculus.

Integrating $-4x$

Now, let's focus on integrating the term $-4x$. This is another straightforward application of the power rule of integration, but it's crucial to handle the negative sign and the constant coefficient correctly. As before, the power rule states that the integral of $x^n$ with respect to $x$ is $(x^{n+1})/(n+1)$, where $n$ is any real number except -1. In this case, we have $-4x$, which can be rewritten as $-4x^1$. This makes it clear that the exponent of $x$ is 1. The first step in integrating $-4x$ is to recognize the constant multiple, which is -4. Similar to the previous example, we can use the constant multiple rule to pull -4 out of the integral. This simplifies the problem, allowing us to focus on integrating $x^1$ first and then multiplying the result by -4. Applying the power rule to $x^1$, we increase the exponent by 1, making it 2, and then divide by the new exponent, which is 2. Therefore, the integral of $x^1$ is $(x^2)/2$. Next, we multiply this result by the constant multiple -4 that we initially factored out. This gives us $-4 * (x^2)/2$, which simplifies to $-2x^2$. To ensure the correctness of our integration, we can differentiate $-2x^2$ with respect to $x$. Using the power rule for differentiation, we bring down the exponent 2 and multiply it by the coefficient -2, giving us -4. We then decrease the exponent by 1, making it 1. Thus, the derivative of $-2x^2$ is $-4x$, which is the original term we started with. This confirms that our integration was performed correctly. This detailed explanation of integrating $-4x$ highlights the importance of handling constant multiples and negative signs with care. It reinforces the application of the power rule and emphasizes the relationship between integration and differentiation as inverse processes. By understanding this process, we can confidently tackle similar integration problems involving polynomial terms.

Integrating $\sin x$

Integrating the trigonometric function $\sin x$ requires recalling a fundamental result from integral calculus. The integral of $\sin x$ with respect to $x$ is $-\cos x$. This result is a standard one, often memorized by students of calculus, but it's also essential to understand where it comes from. The integration of $\sin x$ is based on the reverse process of differentiation. We know that the derivative of $\cos x$ with respect to $x$ is $-\sin x$. Therefore, the antiderivative of $-\sin x$ is $\cos x$. However, we are trying to find the antiderivative of $\sin x$, not $-\sin x$. To account for this, we multiply both sides of the equation by -1, which gives us that the antiderivative of $\sin x$ is $-\cos x$. This fundamental relationship between $\sin x$ and $-\cos x$ is crucial in many areas of mathematics and physics. When integrating $\sin x$, it is essential to remember the negative sign in front of the $\cos x$. A common mistake is to forget this negative sign, leading to an incorrect result. To verify that our integration is correct, we can differentiate $-\cos x$ with respect to $x$. The derivative of $-\cos x$ is $-(-\sin x)$, which simplifies to $\sin x$, the original function we started with. This confirms that our integration was performed correctly. This process underscores the close relationship between integration and differentiation, reinforcing the idea that integration is the reverse process of differentiation. The integral of $\sin x$ as $-\cos x$ is a core concept in calculus, and understanding this result thoroughly is vital for tackling more complex integration problems involving trigonometric functions. By remembering this fundamental result and understanding its derivation, we can confidently integrate $\sin x$ and apply this knowledge to various mathematical contexts.

Combining the Results and Adding the Constant of Integration

After integrating each term individually, we now combine the results to obtain the complete integral of the original expression, $3x^2 - 4x + \sin x$. We found that the integral of $3x^2$ is $x^3$, the integral of $-4x$ is $-2x^2$, and the integral of $\sin x$ is $-\cos x$. Combining these, we have $x^3 - 2x^2 - \cos x$. However, this is not the final answer. A crucial step in indefinite integration is adding the constant of integration, denoted by $C$. The constant of integration arises from the fact that the derivative of a constant is zero. This means that when we find an antiderivative, there are infinitely many possible constants that could be added to it without changing its derivative. For example, the derivative of $x^3$ is $3x^2$, but so is the derivative of $x^3 + 1$, $x^3 - 5$, or $x^3 + C$, where $C$ is any constant. Therefore, when we find an indefinite integral, we must include the constant of integration to represent all possible antiderivatives. Including the constant of integration is not just a formality; it's a fundamental aspect of indefinite integration. Failing to include it means we are only finding one particular antiderivative, not the general antiderivative. So, the final result of integrating $3x^2 - 4x + \sin x$ is $x^3 - 2x^2 - \cos x + C$. This expression represents the family of functions whose derivatives are $3x^2 - 4x + \sin x$. To summarize, we have meticulously integrated each term of the given expression, combined the results, and importantly, added the constant of integration. This ensures that our answer is complete and accurate, representing the general antiderivative of the given function.

Final Answer

Therefore, the final answer to the integral of $3x^2 - 4x + \sin x$ is: $ \int (3x^2 - 4x + \sin x) dx = x^3 - 2x^2 - \cos x + C $ This comprehensive result encapsulates the integration of each term, utilizing the power rule for the polynomial terms and the standard integral for the trigonometric term. The inclusion of the constant of integration, $C$, is paramount, as it signifies the infinite family of antiderivatives that could satisfy the original integral. Understanding this process is crucial for mastering integral calculus and its applications in various scientific and engineering fields. The journey through this integration problem has highlighted several key concepts. We started by dissecting the problem into manageable parts, integrating each term individually using the appropriate rules. This approach emphasizes the importance of breaking down complex problems into simpler components. We applied the power rule for integration to the polynomial terms, a fundamental technique for handling functions of the form $x^n$. We also recalled the standard integral of $\sin x$, which is $-\cos x$. The constant multiple rule allowed us to handle coefficients efficiently, and the concept of indefinite integration was reinforced by the crucial addition of the constant of integration, $C$. This final answer not only provides the solution to the given integral but also serves as a testament to the power of calculus in finding antiderivatives and understanding the relationships between functions and their integrals. By carefully following each step and understanding the underlying principles, we can confidently tackle a wide range of integration problems.