Solving ∫01 (x - 1)9 Dx A Comprehensive Guide To Substitution

Hey guys! Let's dive into the fascinating world of definite integrals, specifically how we can use the substitution technique to solve them. Today, we're tackling a classic problem that might seem a bit daunting at first, but trust me, it's totally manageable once we break it down. We're going to explore the integral ${\int_0^1 (x - 1)^9 \, dx}$ and see how the magic of variable substitution can lead us to the right answer. So, grab your thinking caps, and let's get started!

Understanding the Substitution Technique

The substitution technique, also known as u-substitution, is a powerful tool in calculus that allows us to simplify complex integrals by changing the variable of integration. The main idea behind it is to identify a part of the integrand (the function being integrated) and substitute it with a new variable, often denoted as 'u'. This substitution, when done correctly, can transform the integral into a simpler form that we can easily evaluate.

When to Use Substitution

So, when should you reach for the substitution technique? A good rule of thumb is to consider it when you see a function and its derivative (or a constant multiple of its derivative) within the integral. For example, if you have an integral involving ${\sin(x^2)}$ and ${x}$, the substitution ${u = x^2}$ might be a good idea because the derivative of ${x^2}$ is ${2x}$, which is a multiple of ${x}$. Another scenario is when you have a composite function raised to a power, like in our case, where we have ${(x - 1)^9}$. The substitution technique is invaluable in these cases because it can help you unravel the layers of the composite function.

The Steps Involved

Now, let's walk through the general steps of the substitution technique:

1. Identify the 'u': Choose a suitable expression within the integrand to be your 'u'. This is often the most crucial step and requires a bit of practice and intuition. Look for composite functions or expressions whose derivatives are also present in the integral.
2. Find du: Calculate the derivative of 'u' with respect to 'x', denoted as ${\frac{du}{dx}}$, and then solve for ${du}$.
3. Substitute: Replace the original expression in the integral with 'u' and replace ${dx}$ with the expression you found for ${du}$ in terms of ${dx}$.
4. Evaluate the Integral: Evaluate the new integral in terms of 'u'. This should be a simpler integral than the original one.
5. Substitute Back: Replace 'u' with its original expression in terms of 'x'.
6. Evaluate the Definite Integral: If you're dealing with a definite integral (an integral with limits of integration), you have two options:
   • Option 1: Change the limits of integration to be in terms of 'u' and evaluate the integral with the new limits.
   • Option 2: Substitute back to 'x' as in step 5 and evaluate the integral with the original limits.

Let's see how this plays out in our specific problem.

Applying Substitution to Our Problem: ${\int_0^1 (x - 1)^9 \, dx}$

Alright, let's get our hands dirty and apply the substitution technique to the integral ${\int_0^1 (x - 1)^9 \, dx}$. Remember, the goal is to make this integral easier to handle. Looking at the integrand, ${(x - 1)^9}$, we can see a composite function – something raised to a power. This is a classic sign that substitution might be our friend.

Step 1: Identifying 'u'

The most natural choice for 'u' here is the expression inside the parentheses, which is ${x - 1}$. So, let's set: ${u = x - 1}$

This seems like a good choice because it simplifies the integrand's structure. Instead of dealing with ${(x - 1)^9}$, we'll soon have something like ${u^9}$, which is much easier to integrate.

Step 2: Finding du

Next, we need to find ${du}$, the derivative of 'u' with respect to 'x'. Taking the derivative of ${u = x - 1}$, we get: ${\frac{du}{dx} = 1}$

This is wonderfully simple! Now, solving for ${du}$, we have: ${du = dx}$

This tells us that ${du}$ and ${dx}$ are directly interchangeable in our integral. This is a common and convenient outcome in many substitution problems.

Step 3: Substituting

Now comes the fun part – the substitution itself! We replace ${(x - 1)}$ with 'u' and ${dx}$ with ${du}$ in the original integral. This gives us: ${\int_0^1 (x - 1)^9 \, dx = \int u^9 \, du}$

But hold on a second! We're not quite done with the substitution yet. We have a definite integral, which means we have limits of integration (0 and 1 in this case). These limits are in terms of 'x', but our new integral is in terms of 'u'. We have two choices here:

• Option 1: Change the limits of integration to be in terms of 'u'.
• Option 2: Evaluate the integral in terms of 'u', then substitute back to 'x' and use the original limits.

Let's go with Option 1 for this example, as it's often the more efficient approach. To change the limits, we need to find the values of 'u' that correspond to ${x = 0}$ and ${x = 1}$.

When ${x = 0}$, we have: ${u = 0 - 1 = -1}$

So, the new lower limit of integration is -1.

When ${x = 1}$, we have: ${u = 1 - 1 = 0}$

So, the new upper limit of integration is 0.

Now our integral looks like this: ${\int_0^1 (x - 1)^9 \, dx = \int_{-1}^0 u^9 \, du}$

Notice how the limits have changed to reflect the new variable 'u'. This is a crucial step to ensure we get the correct answer for the definite integral.

Step 4: Evaluating the Integral

Now we have a much simpler integral to evaluate: ${\int_{-1}^0 u^9 \, du}$. This is a straightforward power rule integration. Recall that the power rule states: ${\int x^n \, dx = \frac{x^{n+1}}{n+1} + C}$

Applying this to our integral, we get: ${\int_{-1}^0 u^9 \, du = \left[ \frac{u^{10}}{10} \right]_{-1}^0}$

Notice that we don't need the constant of integration 'C' because we're dealing with a definite integral. Now we just need to evaluate the expression at the upper and lower limits and subtract.

Step 5: Evaluating the Definite Integral

Plugging in the limits of integration, we have: ${\left[ \frac{u^{10}}{10} \right]_{-1}^0 = \frac{0^{10}}{10} - \frac{(-1)^{10}}{10}}$

Simplifying this, we get: ${= 0 - \frac{1}{10} = -\frac{1}{10}}$

So, the value of the definite integral is ${-\frac{1}{10}}$.

The Final Answer and Conclusion

Therefore, the definite integral ${\int_0^1 (x - 1)^9 \, dx}$ evaluates to ${-\frac{1}{10}}$. Looking at the options provided, none of them exactly match our answer. However, option (A) is 1/10. It seems there might be a sign error in the options or the original question, as our calculation clearly shows the answer to be negative. It's essential to double-check your work and the given options when faced with such discrepancies.

In conclusion, we've successfully used the substitution technique to evaluate a definite integral. We saw how choosing the right 'u', finding ${du}$, and changing the limits of integration (or substituting back to 'x') are crucial steps in the process. This technique is a cornerstone of integral calculus, and mastering it will open doors to solving a wide range of integration problems. Keep practicing, and you'll become a substitution pro in no time!

Remember, the key to mastering integration techniques is practice, practice, practice! Don't be afraid to tackle different problems and experiment with different substitutions. The more you work with these concepts, the more intuitive they'll become. And hey, if you ever get stuck, don't hesitate to reach out for help. There are tons of resources available online and in textbooks, and your instructors and classmates are always there to lend a hand. Keep up the great work, guys, and happy integrating!