Problem 33 - College Algebra and Calculus: An Applied Approach

Question

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int x \sqrt{x-1} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(x\sqrt{x-1}\) dx is \( [(2/5)(x - 1)^{2.5} + (2/3)(x - 1)^{1.5}] + C \).

Step 1: Rewrite the Function

Rewrite the function in the following way: \( \int x(x - 1)^{0.5} dx\). This simplifies the function and makes it easier to solve.

Step 2: U-substitution

Substitute \(u = x - 1\), hence \(du = dx\), and \(x = u + 1\). Substitute these values into the integral to get: \( \int (u + 1)(u)^{0.5} du\).

Step 3: Distribute and split the integral

Apply the integral rules including the distribution of the \(u^{0.5}\), giving \( \int (u^{1.5} + u^{0.5}) du\). Now, split this into two separate integrals: \( \int u^{1.5} du + \int u^{0.5} du\).

Step 4: Use the Power Rule

Apply the power rule of integration, resulting in: \( [(2/5)u^{2.5} + (2/3)u^{1.5}] + C \). Here, 'C' represents the constant of integration.

Step 5: Back-substitution

Substitute back \(u = x - 1\) to give the answer in terms of x: \( [(2/5)(x - 1)^{2.5} + (2/3)(x - 1)^{1.5}] + C \).

Key Concepts

Understanding U-substitution

The method of u-substitution is a powerful technique used to simplify the integration of functions. In this context, it's often employed to deal with composite functions, especially when their integration isn't straightforward.
The idea is to substitute part of the integrand with a single variable, typically "u," to simplify the expression. Let's look at our exercise:

• The original integrand: \( x \sqrt{x-1} \)
• Set \( u = x - 1 \), meaning the derivative \( du = dx \)
• Thus, \( x = u + 1 \)

By making this substitution, the integral becomes \( \int (u + 1)u^{0.5} \, du \), which is much easier to handle than the original expression. This significant simplification is what makes u-substitution a key method in calculus.

Power Rule of Integration

The power rule of integration is an essential tool that helps integrate expressions with terms raised to a power. It provides a straightforward formula:\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]where \( n eq -1 \) and \( C \) is the constant of integration.
In this problem, after the u-substitution, we split the integral into two terms: \( \int u^{1.5} \, du + \int u^{0.5} \, du \). Applying the power rule to each:

• For \( \int u^{1.5} \, du \), we get \( \frac{2}{5} u^{2.5} \)
• For \( \int u^{0.5} \, du \), we get \( \frac{2}{3} u^{1.5} \)

Using the power rule simplifies each term effectively, rendering a complex expression into manageable parts. This is why it's vital for solving integrals.

Importance of the Constant of Integration

In indefinite integrals, the constant of integration, represented by "C," holds critical importance. It embodies the family of all antiderivatives of a function since indefinite integrals indicate a range of possible solutions.
When we integrate, we essentially reverse the differentiation process, which means the result includes all possible vertical shifts of the function derived from the constant "C." Consider these points:

• The inclusion of "C" ensures the solution covers all contexts.
• Every additional constant changes the curve's position vertically, without altering its shape.

Hence, in any indefinite integral, always add "C" to reflect the complete solution, accounting for an infinite number of parallel curves.