Problem 262 - Calculus Volume 1

Question

In the following exercises, find the antiderivative using the indicated substitution. $$ \int(x-1)^{5} d x ; u=x-1 $$

Step-by-Step Solution

Answer

\( \frac{(x-1)^6}{6} + C \).

Step 1: Identify Substitution

We are given the substitution variable, where \( u = x - 1 \). This means that \( du = dx \) because the derivative of \( x - 1 \) with respect to \( x \) is 1.

Step 2: Substitute in the Integral

Replace \( x - 1 \) with \( u \) in the integral. The differential \( dx \) is replaced with \( du \) as derived. The integral becomes: \( \int u^5 \, du \).

Step 3: Find the Antiderivative

The antiderivative of \( u^5 \) with respect to \( u \) is \( \frac{u^{6}}{6} + C \), where \( C \) is the constant of integration.

Step 4: Back-Substitute the Variable

Since \( u = x - 1 \), substitute back to get the antiderivative in terms of \( x \). This gives us: \( \frac{(x-1)^{6}}{6} + C \).

Key Concepts

Understanding Integration by Substitution

Integration by substitution is a method used to find integrals more easily. It's akin to undoing the chain rule from differentiation. This technique is particularly useful when you have an integral that involves a composite function. The general idea is to simplify the integral by substituting part of it with a new variable.
This new variable should represent a simple expression inside the function being integrated.To begin with substitution:

• Identify a part of the integral that, when substituted, becomes simpler.
• Define your substitution. For instance, if you have an integral like \( \int (x-1)^5 \, dx \), let \( u = x - 1 \). This makes the expression simpler to integrate.
• Find the differential \( du \), which will replace \( dx \). In our example, since \( u = x - 1 \), \( du = dx \).
• Replace the identified part and the differential with the new variables. This transforms the integral into a simpler form, \( \int u^5 \, du \), which is easier to integrate.
• Finally, after finding the antiderivative, substitute back to the original variable.

Integration by substitution reduces tricky integrals into basic forms, making calculations more manageable. It's a handy tool that matches well with problems involving nested functions or expressions.

Exploring Integrals

Integrals are fundamental in calculus, representing the accumulation of quantities. Consider them as the opposite of derivatives, used to find functions when we know how their rate of change looks. Think of integrals as calculating the area under a curve—in a two-dimensional graph, it’s the area between the curve and the x-axis.
There are two main types of integrals:

• Indefinite Integrals: These are functions that represent entire families of curves, noted usually as \( F(x) + C \). The \( C \) is the constant of integration because derivates can remove constant terms.
• Definite Integrals: These compute the actual area under a curve between two specific values, providing a numerical outcome.

In this particular exercise, we dealt with an indefinite integral, searching for a general formula that describes the overall behavior of the original function. Understanding integrals demands practice and familiarity, but knowing their role and purpose makes solving calculus problems more intuitive.

Solving Calculus Problems Effectively

Calculus problems often seem daunting because of their abstract nature. Yet, like puzzles, they become exciting once you approach them strategically. Here's how you can handle calculus problems without getting overwhelmed:

• Understand the Problem: Grasp the fundamental question. Are you asked to find a rate of change (derivative) or an accumulated value (integral)? Knowing this sets your approach.
• Break it Down: Divide the problem into smaller steps. For example, each integral might require substitution or different algebraic manipulations. Repeat smaller tasks you understand well.
• Use Substitution Wisely: As illustrated in the original exercise, substitution transforms a tricky integral into an easier one. It's crucial to choose substitutions that simplify the integral calculation.
• Check Your Work: After solving, go back. Derive your integral solution if possible, and ensure it returns to the original function form. This confirms correctness.

Patience and practice strengthen your calculus skills. Remember, it's not just about arriving at the solution but understanding the journey there. With repeated exposure to various calculus problems, forming a methodical approach becomes second nature.