Integrating -4x^4 - 4/x - 5/x^6 - 2√x: A Step-by-Step Guide

Aug 6, 2025 by Sebastian Müller 60 views

Unraveling the Integral of (-4x^4 - 4/x - 5/x^6 - 2√x) dx: A Comprehensive Guide

Hey guys! Today, we're diving deep into the fascinating world of calculus to tackle a rather intriguing integral problem. We're going to break down the process of finding the integral of the function (-4x^4 - 4/x - 5/x^6 - 2√x) dx. Trust me, it might look a bit intimidating at first glance, but with a step-by-step approach and a sprinkle of calculus magic, we'll conquer this beast together. So, buckle up and let's get started!

Understanding the Integral

Before we jump into the nitty-gritty, let's take a moment to understand what an integral actually represents. In simple terms, an integral is the reverse operation of differentiation. While differentiation helps us find the rate of change of a function, integration helps us find the area under the curve of a function. Think of it like this: differentiation is like zooming in on a curve to see its slope at a particular point, while integration is like zooming out to see the cumulative effect of the function over an interval.

The integral we're dealing with, ∫(-4x^4 - 4/x - 5/x^6 - 2√x) dx, is a definite integral. This means we're looking for a general function whose derivative is equal to the expression inside the integral. This function, plus a constant of integration (C), represents the family of all functions that satisfy this condition. The constant of integration is crucial because the derivative of a constant is always zero, so there are infinitely many functions that could have the same derivative.

Now, let's talk about the components of our integral. We have four terms: -4x^4, -4/x, -5/x^6, and -2√x. Each of these terms represents a different function, and we'll need to apply the rules of integration to each one individually. This is where the power of calculus shines – we can break down complex problems into smaller, more manageable parts. So, let's dive into the first term and see how we can integrate it.

Breaking Down the Integral Term by Term

Now, let's dissect this integral piece by piece. Our integral is ∫(-4x^4 - 4/x - 5/x^6 - 2√x) dx. We can break this down into four separate integrals using the linearity property of integrals, which states that the integral of a sum (or difference) of functions is equal to the sum (or difference) of their integrals. This gives us:

∫(-4x^4) dx - ∫(4/x) dx - ∫(5/x^6) dx - ∫(2√x) dx

This breakdown makes the problem much more approachable. We can now focus on integrating each term individually and then combine the results. This is a common strategy in calculus – breaking down complex problems into simpler ones. Each integral now looks more manageable, and we can apply the appropriate integration rules to solve them.

Integrating -4x^4

The first term we'll tackle is -4x^4. This is a power function, and we can use the power rule of integration to find its integral. The power rule states that:

∫x^n dx = (x^(n+1)) / (n+1) + C, where n ≠ -1

In our case, we have -4x^4, so n = 4. Applying the power rule, we get:

∫(-4x^4) dx = -4 ∫x^4 dx = -4 * (x^(4+1)) / (4+1) + C = -4 * (x^5) / 5 + C = -4/5 * x^5 + C

So, the integral of -4x^4 is -4/5 * x^5 + C. Remember, the constant of integration, C, is essential because there are infinitely many functions whose derivative is -4x^4. We've successfully integrated the first term! Let's move on to the next one.

Integrating -4/x

Next up, we have the term -4/x. This might look like a power function, but it's actually a special case. Remember that the power rule for integration doesn't apply when n = -1. In this case, we have -4/x, which can be rewritten as -4 * (1/x). The integral of 1/x is a well-known result:

∫(1/x) dx = ln|x| + C

So, to integrate -4/x, we can use the constant multiple rule, which states that the integral of a constant times a function is equal to the constant times the integral of the function. This gives us:

∫(-4/x) dx = -4 ∫(1/x) dx = -4 * ln|x| + C

The integral of -4/x is -4ln|x| + C. Notice the absolute value sign inside the logarithm. This is important because the logarithm function is only defined for positive values, and we want our integral to be valid for all x ≠ 0. We're making great progress! Let's move on to the third term.

Integrating -5/x^6

Now, let's tackle the term -5/x^6. This might look intimidating, but we can rewrite it as a power function by using the property that 1/x^n = x^(-n). So, -5/x^6 can be written as -5x^(-6). Now we can apply the power rule of integration:

∫x^n dx = (x^(n+1)) / (n+1) + C, where n ≠ -1

In this case, n = -6. Applying the power rule, we get:

∫(-5/x^6) dx = ∫(-5x^(-6)) dx = -5 ∫x^(-6) dx = -5 * (x^(-6+1)) / (-6+1) + C = -5 * (x^(-5)) / (-5) + C = x^(-5) + C

So, the integral of -5/x^6 is x^(-5) + C, which can also be written as 1/x^5 + C. We're cruising through this integral! Only one term left to go.

Integrating -2√x

Finally, let's integrate the term -2√x. To make this easier, we can rewrite the square root as a fractional exponent: √x = x^(1/2). So, our term becomes -2x^(1/2). Now we can apply the power rule of integration:

∫x^n dx = (x^(n+1)) / (n+1) + C, where n ≠ -1

In this case, n = 1/2. Applying the power rule, we get:

∫(-2√x) dx = ∫(-2x^(1/2)) dx = -2 ∫x^(1/2) dx = -2 * (x^(1/2 + 1)) / (1/2 + 1) + C = -2 * (x^(3/2)) / (3/2) + C = -2 * (2/3) * x^(3/2) + C = -4/3 * x^(3/2) + C

So, the integral of -2√x is -4/3 * x^(3/2) + C. We've successfully integrated all four terms! Now, let's put it all together.

Combining the Results

We've successfully integrated each term of our original expression. Now, we need to combine the results to find the complete integral. Remember, we broke down the integral into four parts:

∫(-4x^4 - 4/x - 5/x^6 - 2√x) dx = ∫(-4x^4) dx - ∫(4/x) dx - ∫(5/x^6) dx - ∫(2√x) dx

We found the integrals of each term to be:

∫(-4x^4) dx = -4/5 * x^5 + C
∫(-4/x) dx = -4ln|x| + C
∫(-5/x^6) dx = 1/x^5 + C
∫(-2√x) dx = -4/3 * x^(3/2) + C

Now, let's add these results together. Since we have a constant of integration, C, for each term, we can combine them into a single constant of integration. This gives us:

∫(-4x^4 - 4/x - 5/x^6 - 2√x) dx = -4/5 * x^5 - 4ln|x| + 1/x^5 - 4/3 * x^(3/2) + C

And there you have it! We've successfully found the integral of the given expression. It might have seemed daunting at first, but by breaking it down into smaller parts and applying the rules of integration, we were able to solve it. Give yourself a pat on the back – you've earned it!

Final Answer

The final answer to the integral ∫(-4x^4 - 4/x - 5/x^6 - 2√x) dx is:

-4/5 * x^5 - 4ln|x| + 1/x^5 - 4/3 * x^(3/2) + C

Remember, the constant of integration, C, is crucial because it represents the family of all functions that have the same derivative as the expression inside the integral. We've successfully navigated the world of calculus and found the integral! I hope this comprehensive guide has helped you understand the process and build your confidence in tackling similar problems. Keep practicing, and you'll become a calculus pro in no time!