Problem 34 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The indefinite integral of \( \frac{x}{\sqrt{x-1}} \) is \( \frac{2(x - 1)^{(3/2)}}{3} + 2(x - 1)^{1/2} + C \)

Step 1: Identify The Form

Observe that the function is of the form \( \frac{x}{\sqrt{x-1}} \). This form suggests a substitution might be helpful.

Step 2: Substitute

Let \( u = x - 1 \). Then, \( du = dx \) and \( x = u + 1 \)

Step 3: Rewrite the Integral

Now, the integral can be rewritten in terms of \( u \). So, the integral \( \int \frac{x}{\sqrt{x-1}} dx \) becomes \( \int \frac{u + 1}{\sqrt{u}} du \)

Step 4: Split the Integral

The integral can be split into two simpler integrals: \( \int \frac{u}{\sqrt{u}} du + \int \frac{1}{\sqrt{u}} du \)

Step 5: Simplify the Integrals

The integrals can be simplified further to \( \int u^{1/2-1} du + \int u^{-1/2} du \)

Step 6: Integrate

Integrating using the power rule \( \int u^n du = \frac{u^{n+1}}{n+1} \) gives \( \frac{2u^{3/2}}{3} + 2u^{1/2} \)

Step 7: Substitute Back

Finally, substitute back \( u = x - 1 \) into the integral to get \( \frac{2(x - 1)^{(3/2)}}{3} + 2(x - 1)^{1/2} + C \), where \( C \) is the constant of integration.

Key Concepts

Integration by Substitution

When faced with a complex integral, one technique that can simplify the process is integration by substitution, often referred to as u-substitution. This method involves changing the variable of integration from one to another that converts the integral into a more manageable form.

For example, consider the provided exercise that requires us to solve the integral of the function \( \frac{x}{\sqrt{x-1}} \). The function under the integral sign suggests that substitution may be beneficial. Here's how:

• Identify a part of the integral as \( u \), which in this case is \( u = x - 1 \).
• Differentiate \( u \) with respect to \( x \) to find \( du \) which equals \( dx \) in this example.
• Replace \( x \) in the integral with \( u + 1 \) and \( dx \) with \( du \) to express the integral solely in terms of \( u \).

After substitution, you can often split the integral into simpler parts to integrate more easily, just as was done in the exercise.
Understanding when and how to apply integration by substitution can greatly facilitate the solving of indefinite integrals and is a fundamental skill in calculus.

Power Rule for Integration

The power rule for integration is one of the most straightforward and commonly used rules for finding antiderivatives. It comes into play when you need to integrate a function of the form \( x^n \). The rule states that:
\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C,\) where \( n \) is any real number except -1, and \( C \) represents the constant of integration.

In our exercise, after applying substitution and splitting the integral, we were left with simpler integrals of the form \( u^{n} \), to which we could apply the power rule.

• For \( u^{1/2-1}\) or \(u^{-1/2}\), the power rule gives us \( 2u^{1/2} \).
• Similarly, for \( u^{-1/2} \), it provides \( 2u^{1/2} \).

By applying the power rule correctly, we effectively simplified the integrals that arose from the substitution process. Mastering the power rule is essential, as it not only saves time but also boosts your problem-solving technique arsenal.

Antiderivatives

An antiderivative of a function is another function whose derivative is the original function. In basic terms, it's the reverse process of differentiating. While derivatives represent rates of change, antiderivatives can be thought of as accumulation functions. They are vital to calculus, especially when determining the area under a curve.

For example, if we want to find an antiderivative of \( f(x) = x \), we look for a function \( F(x) \) such that \( F'(x) = x \). Employing the power rule, we find that \( F(x) = \frac{x^2}{2} + C \) is such a function.

In the integral from our exercise, \( \int \frac{x}{\sqrt{x-1}} dx \), the goal is to find an antiderivative. The steps involved using substitution and the power rule lead to the antiderivative of the given function, which happened to be \( \frac{2(x - 1)^{(3/2)}}{3} + 2(x - 1)^{1/2} + C \). Here, \( C \) signifies the constant of integration, which appears because indefinite integrals represent families of functions.
Grasping the concept of antiderivatives and the techniques to find them is crucial for many areas in mathematics and physics, such as calculating work done by a force or finding the position of an object given its velocity.