Solving The Integral Of (x + 4sin(x) - 2) Dx

Hey math whizzes and calculus adventurers! Today, we're diving deep into the fascinating world of integration to tackle a specific problem: finding the integral of $(x + 4\sin(x) - 2)dx$. If you're looking to master integration techniques, you've come to the right place, guys. We'll break down this integral step-by-step, making sure you understand every bit of it. So, grab your notebooks and let's get this calculus party started!

Understanding the Basics of Integration

Before we jump into our specific integral, let's quickly refresh what integration is all about. In simple terms, integration is the reverse process of differentiation. While differentiation helps us find the rate of change of a function, integration helps us find the accumulated change or the area under a curve. When we talk about the integral of a function, we're essentially looking for a function whose derivative is the original function. This is also known as finding the antiderivative. The symbol '∫' represents the integral, and 'dx' at the end tells us that we are integrating with respect to the variable x. For our problem, we need to find $\int (x + 4\sin(x) - 2)dx$. This involves applying the fundamental rules of integration to each term within the parentheses. It's like unboxing a present; you deal with each item inside separately before you can appreciate the whole gift. So, let's get ready to unbox this integral!

Breaking Down the Integral: Term by Term

Our integral, $\int (x + 4\sin(x) - 2)dx$, is composed of three distinct terms: $x$, $4\sin(x)$, and $-2$. The beauty of integration is that it's a linear operation. This means we can integrate each term separately and then sum up the results. This property, known as the sum and difference rule for integration, is super handy. It allows us to simplify complex integrals into a series of simpler ones. So, we can rewrite our integral as:

$\int x dx + \int 4\sin(x) dx - \int 2 dx$

Now, let's tackle each of these integrals one by one. This approach makes the entire process much more manageable and less intimidating. Think of it as solving a puzzle; you find the right place for each piece before you see the complete picture. We're going to apply some fundamental integration rules that you've probably encountered in your calculus journey. These rules are the building blocks for solving a vast array of integration problems, and mastering them is key to unlocking more advanced calculus concepts. Let's dive into the first term, the simple yet crucial power of x.

Integrating the Power of 'x': The Power Rule

The first term we need to integrate is $\int x dx$. This is a classic example of using the power rule for integration. The power rule states that for any real number $n \neq -1$, the integral of $x^n$ is given by:

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$

In our case, the term is just $x$, which can be thought of as $x^1$. So, $n=1$. Applying the power rule, we get:

$\int x^1 dx = \frac{x^{1+1}}{1+1} + C_1 = \frac{x^2}{2} + C_1$

Here, $C_1$ is the constant of integration. We include this constant because the derivative of any constant is zero. So, when we find an antiderivative, there are infinitely many possibilities, differing only by a constant. We'll combine all these constants at the end into a single one.

This rule is incredibly fundamental, guys, and it forms the basis for integrating most polynomial terms. Remember, the key is to increase the exponent by one and then divide by the new exponent. It's straightforward and immensely powerful. Make sure you've got this one locked down, as you'll be using it all the time in your calculus studies. It’s one of those core concepts that just keeps showing up, no matter how complex the problem gets. So, we've successfully integrated the $x$ term. High five! Now, let's move on to the next part of our integral, which involves a trigonometric function. Get ready!

Integrating Trigonometric Functions: The Sine Rule

Next up, we have the term $\int 4\sin(x) dx$. This involves integrating a trigonometric function, specifically the sine function, multiplied by a constant. First, let's address the constant '4'. Just like in integration, constants can be pulled out of the integral sign. So, we have:

$\int 4\sin(x) dx = 4 \int \sin(x) dx$

Now, we need to find the integral of $\sin(x)$. Recall from differentiation that the derivative of $\cos(x)$ is $-\sin(x)$. Therefore, the integral of $\sin(x)$ is $-\cos(x)$. So, we have:

$\int \sin(x) dx = -\cos(x) + C_2$

Putting it all together, the integral of $4\sin(x)$ is:

$4 \int \sin(x) dx = 4(-\cos(x) + C_2) = -4\cos(x) + 4C_2$

Again, $4C_2$ is just another constant. We'll combine it with other constants later. So, the integral of $4\sin(x)$ is $-4\cos(x)$ (plus a constant, of course).

This is a crucial rule to remember for integrating trigonometric functions. The integral of $\cos(x)$ is $\sin(x)$, and the integral of $\sin(x)$ is $-\cos(x)$. It’s a neat little relationship that keeps calculus moving. Always remember that sign change when going from differentiation to integration for sine and cosine. It’s a common stumbling block for many, so pay close attention to it! We’re doing great, guys. Just one more term to go, and then we’ll have our complete solution. Let’s tackle that constant term.

Integrating Constant Terms

The final term in our integral is $\int -2 dx$. Integrating a constant is perhaps the simplest rule of all. For any constant $k$, the integral of $k$ with respect to $x$ is simply $kx$ plus a constant of integration.

$\int k dx = kx + C_3$

In our case, $k = -2$. So, the integral is:

$\int -2 dx = -2x + C_3$

That was pretty straightforward, right? It’s like finding the area of a rectangle; you just multiply the base (which is x, or the change in x) by the height (which is the constant value). This rule is fundamental because many functions can be expressed as sums or differences of terms involving constants, powers of x, and trigonometric functions. Understanding how to integrate each of these basic forms unlocks the ability to integrate a much wider variety of functions. So, we've now integrated all three parts of our original expression. It's time to put it all together!

Combining the Results: The Final Answer

We've successfully integrated each term of our original expression $\int (x + 4\sin(x) - 2)dx$. Now, we just need to combine the results and add a single constant of integration. Remember, the sum and difference rule allowed us to break it down, and now we put it back together:

$\int x dx = \frac{x^2}{2} + C_1$

$\int 4\sin(x) dx = -4\cos(x) + C_2$

$\int -2 dx = -2x + C_3$

Combining these, we get:

$\frac{x^2}{2} + C_1 - 4\cos(x) + C_2 - 2x + C_3$

We can group all the constants ($C_1$, $C_2$, and $C_3$) into a single arbitrary constant, let's call it $C$. Since the sum of arbitrary constants is just another arbitrary constant, we can write:

$C = C_1 + C_2 + C_3$

So, the final integrated expression is:

$\frac{x^2}{2} - 4\cos(x) - 2x + C$

And there you have it, guys! The solution to our integral. This is the indefinite integral, meaning it represents a family of functions, each differing by a constant value $C$. If we had a definite integral (with limits of integration), we would plug in those limits to find a specific numerical value. But for indefinite integrals, this is our final destination.

Verification: Differentiating the Result

To be absolutely sure we've got the right answer, we can always perform a quick check by differentiating our result. If we differentiate $\frac{x^2}{2} - 4\cos(x) - 2x + C$ with respect to $x$, we should get back our original function, $x + 4\sin(x) - 2$. Let's do it:

$\frac{d}{dx}(\frac{x^2}{2} - 4\cos(x) - 2x + C)$

Applying the differentiation rules term by term:

• Derivative of $\frac{x^2}{2}$: Using the power rule for differentiation ($\frac{d}{dx}(x^n) = nx^{n-1}$), we get $\frac{d}{dx}(\frac{1}{2}x^2) = \frac{1}{2} \cdot 2x^{2-1} = x$. This matches our first term!
• Derivative of $-4\cos(x)$: The derivative of $\cos(x)$ is $-\sin(x)$. So, $\frac{d}{dx}(-4\cos(x)) = -4 \cdot (-\sin(x)) = 4\sin(x)$. This matches our second term!
• Derivative of $-2x$: The derivative of $-2x$ is simply $-2$. This matches our third term!
• Derivative of $C$: The derivative of any constant is $0$. So, $\frac{d}{dx}(C) = 0$.

Adding these derivatives together:

$x + 4\sin(x) - 2 + 0 = x + 4\sin(x) - 2$

This is exactly our original integrand! This verification step is super important, especially when you're learning. It builds confidence in your answers and helps you catch any silly mistakes. So, yes, our integration was correct, guys!

Conclusion

We successfully found the integral of $(x + 4\sin(x) - 2)dx$. By breaking the integral into simpler parts, applying the power rule, the trigonometric integral rule for sine, and the constant rule, and then combining the results, we arrived at the solution $\frac{x^2}{2} - 4\cos(x) - 2x + C$. Remember, the key takeaways are the linearity of integration (you can integrate term by term) and the fundamental rules for powers, trigonometric functions, and constants. Always remember to add that constant of integration, $C$, for indefinite integrals. And don't forget to verify your answers by differentiating! Keep practicing, and you'll become a calculus pro in no time. Happy integrating!