Calculate Integral: ∫(1 To 2) Dx/x^4. Step-by-Step!

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Hey guys! Today, we're diving into a fun little calculus problem. We're going to calculate the definite integral of ∫12dxx4\int_{1}^{2} \frac{dx}{x^4}. Don't worry; I'll walk you through each step so it's super easy to follow. Let's get started!

Understanding the Integral

So, when you first look at ∫12dxx4\int_{1}^{2} \frac{dx}{x^4}, it might seem a bit intimidating. But trust me, it's totally manageable. The integral symbol ∫\int basically means we're finding the area under the curve of the function 1x4\frac{1}{x^4} between the limits 1 and 2. These limits tell us where to start and stop calculating the area.

Before we jump into the actual calculation, let's make sure we're all on the same page with some basic integral rules. The main rule we'll use here is the power rule for integration, which states that ∫xndx=xn+1n+1+C\int x^n dx = \frac{x^{n+1}}{n+1} + C, where C is the constant of integration. However, since we're dealing with a definite integral, we won't need to worry about the constant C because it cancels out when we evaluate the integral at the upper and lower limits.

Now, let's rewrite our function 1x4\frac{1}{x^4} as x−4x^{-4}. This makes it easier to apply the power rule. So, our integral becomes ∫12x−4dx\int_{1}^{2} x^{-4} dx. See? Already less scary! Remember, understanding what you're doing is half the battle. This rewritten form highlights that we are integrating a power function, which directly aligns with the power rule we discussed. Recognizing these patterns is super important in calculus. Okay, are you ready to move on and calculate the integral? I bet you are!

Step-by-Step Calculation

Okay, let's get our hands dirty and calculate this integral step by step. Remember, our integral is ∫12x−4dx\int_{1}^{2} x^{-4} dx.

Step 1: Apply the Power Rule

Using the power rule ∫xndx=xn+1n+1\int x^n dx = \frac{x^{n+1}}{n+1}, we get:

∫x−4dx=x−4+1−4+1=x−3−3=−13x3\int x^{-4} dx = \frac{x^{-4+1}}{-4+1} = \frac{x^{-3}}{-3} = -\frac{1}{3x^3}

So, the indefinite integral of x−4x^{-4} is −13x3-\frac{1}{3x^3}.

Step 2: Evaluate at the Upper and Lower Limits

Now we need to evaluate this at the upper limit (2) and the lower limit (1) and subtract the results. This is the fundamental theorem of calculus in action! We write this as:

[−13x3]12=(−13(2)3)−(−13(1)3)\left[-\frac{1}{3x^3}\right]_{1}^{2} = \left(-\frac{1}{3(2)^3}\right) - \left(-\frac{1}{3(1)^3}\right)

Step 3: Simplify

Let's simplify the expression:

(−13(8))−(−13(1))=−124+13\left(-\frac{1}{3(8)}\right) - \left(-\frac{1}{3(1)}\right) = -\frac{1}{24} + \frac{1}{3}

Now, we need to find a common denominator to add these fractions. The common denominator of 24 and 3 is 24. So, we rewrite 13\frac{1}{3} as 824\frac{8}{24}:

−124+824=724-\frac{1}{24} + \frac{8}{24} = \frac{7}{24}

So, the value of the definite integral is 724\frac{7}{24}.

Choosing the Correct Answer

Alright, now that we've done the calculation, let's circle back to the options given and pick the right one:

a) 924\frac{9}{24}

b) 12\frac{1}{2}

c) 724\frac{7}{24}

d) 3154\frac{31}{54}

Based on our calculations, the correct answer is:

c) 724\frac{7}{24}

Woohoo! We got it! You see, even seemingly complex problems can be broken down into simple steps. Understanding each step makes it so much easier. This step-by-step approach not only helps in solving the problem correctly but also boosts your confidence in tackling similar questions in the future.

Common Mistakes to Avoid

Alright, before we wrap up, let’s chat about some common pitfalls students often encounter when tackling integrals like this. Avoiding these mistakes can save you a lot of headaches (and points!).

  1. Forgetting the Power Rule: The power rule is your best friend for integrals of the form ∫xndx\int x^n dx. A very common mistake is to forget to add 1 to the exponent and then divide by the new exponent. Always double-check this step!

  2. Incorrectly Handling Negative Exponents: When you're dealing with negative exponents (like our x−4x^{-4}), it’s easy to make a mistake. Remember that x−4x^{-4} is the same as 1x4\frac{1}{x^4}. Keep track of your signs!

  3. Not Evaluating at the Limits Correctly: This is where many students slip up. Make sure you correctly substitute the upper and lower limits into the antiderivative and subtract in the right order: (value at upper limit) - (value at lower limit). Flipping this order will give you the wrong sign!

  4. Arithmetic Errors: Simple arithmetic mistakes can ruin your whole calculation. Take your time when adding and subtracting fractions, especially when dealing with different denominators. Double-check your arithmetic to avoid these silly errors.

  5. Forgetting the Constant of Integration: While it doesn't affect the final answer in definite integrals, it's crucial to remember the constant of integration (+C) for indefinite integrals. Forgetting it can lead to errors in other contexts.

Practice Makes Perfect

Okay, guys, we've reached the end of our little calculus adventure! Remember, practice makes perfect. The more you work through these problems, the more comfortable and confident you'll become. Calculus might seem tricky at first, but with a bit of patience and a step-by-step approach, you can conquer it! Keep practicing, and you'll be a calculus whiz in no time! If you have any questions, feel free to ask. Happy integrating!