Problem 7 - Calculus of a Single Variable

Question

Use a table of integrals with forms involving the trigonometric functions to find the indefinite integral. \(\int \frac{1}{\sqrt{x}(1-\cos \sqrt{x})} d x\)

Step-by-Step Solution

Answer

The indefinite integral of the given function is \(-\cot(\sqrt{x}/2) + C\).

Step 1: Rewrite using trigonometric identity

We can simplify the given integral by replacing the term \(1-\cos \sqrt{x}\) using the fact that \(1-\cos2x = 2\sin^2x\). As \(\sqrt{x}\) serves as our 'x' here, we rewrite the integral as: \(\int \frac{1}{\sqrt{x} * 2\sin^2(\sqrt{x}/2)} dx\).

Step 2: Perform substitution

Next, we can perform a substitution where we let \(u = \sqrt{x}/2\), hence \(du = \frac{1}{2\sqrt{x}} dx\). Also, \(2du = \frac{1}{\sqrt{x}} dx\), so the integral becomes: \(\int \frac{1}{2\sin^2u} * 2du\). This simplifies to: \(\int \csc^2u du\).

Step 3: Use Table of Integrals

The integrand is now in a standard form found on the table of integrals. The integral of \(\csc^2u\) is \(-\cot(u)\). So we have: \(-\cot(u)\).

Step 4: Back substitution

Lastly, substitute the original variable back in place of 'u' to get: \(-\cot(\sqrt{x}/2) + C\), where 'C' is the constant of integration.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals by substituting trigonometric identities for algebraic expressions. The basic idea is to transform the integral into a form that is easier to evaluate by relating the function to a known trigonometric identity. In the given exercise, we deal with the expression \(1 - \cos \sqrt{x}\). This part of the expression is simplified using a trigonometric identity: \(1 - \cos 2x = 2\sin^2 x\). Here, we reframe our problem so that \(\sqrt{x}\) becomes our variable like \(2x\) in the identity, which is then expressed in terms of sine. This substitution not only makes the integral more manageable but also allows us to use integral tables effectively. By transitioning to trigonometric terms, the integration often becomes straightforward, leading us to the next steps in solving the integral.

Csc Function

The csc, or cosecant function, is the reciprocal of the sine function. It is defined as \(\csc(x) = \frac{1}{\sin(x)}\). The role of the csc function in integration is quite significant, especially when the transformed integrand involves \(\sin(x)\).In step 2 of our solution, we transform the integrand into \(\int \csc^2(u) \, du\). Here, the original expression has been simplified using the identity \(\csc(x) = \frac{1}{\sin(x)}\), which makes here the integral much simpler to evaluate. The importance of understanding the csc function lies in recognizing how transformations using trigonometric identities can simplify complex integrations. Furthermore, recognizing these common trigonometric forms helps us utilize integral tables more efficiently, as seen in this exercise.

Integral Tables

Integral tables are extremely handy tools that list various integral forms and their respective solutions. They serve as a quick reference to solve integrals that match common or complex function forms without having to derive each solution from scratch.In this exercise, after rewriting the integral as \(\int \csc^2(u) \, du\), we refer to an integral table to find its solution. The table states that the integral of \(\csc^2(u)\) is \(-\cot(u)\). This saves us from performing additional calculations, which could potentially involve complex algebraic manipulations.It's essential for students to become familiar with these tables and understand how to use them effectively. By recognizing which table entry corresponds with the transformed integrand, students can efficiently find solutions to a wide range of integral problems, broadening their integration toolset.