Question

Show that with the change of variables \(u=\sqrt{\tan x}\) the integral \(\int \sqrt{\tan x} d x\) can be converted to an integral amenable to partial fractions. Evaluate \(\int_{0}^{\pi / 4} \sqrt{\tan x} d x\).

Short answer

Question: Evaluate the integral \(\int_{0}^{\pi/4} \sqrt{\tan x} dx\) using substitution \(u = \sqrt{\tan x}\). Answer: \(\frac{2}{3}\).

Step-by-step solution

Substitute \(u=\sqrt{\tan x}\)

Let's start with the substitution. Rewrite the given integral in terms of \(u\). Given: \(\int \sqrt{\tan x} dx\), where \(u=\sqrt{\tan x}\). Squaring both sides, we get \(u^2 = \tan x\). Now, we need to find a new differential: \(du/dx\). Taking the derivative of \(u^2\) with respect to \(x\): \(\frac{d(u^2)}{dx} = \frac{d(\tan x)}{dx}\) \(2u \frac{du}{dx} = \sec^2{x}\) Now, solve for \(\frac{du}{dx}\): \(\frac{du}{dx} = \frac{\sec^2{x}}{2u}\) Now we replace \(dx\) with this value in the integral: \(\int \sqrt{\tan x} dx = \int u d\left(\frac{\sec^2{x}}{2u}\right)\)

Simplify and rewrite in terms of \(u\)

Now let's simplify and rewrite this integral in terms of \(u\). We have \(\int u d\left(\frac{\sec^2{x}}{2u}\right) = \int u \cdot \frac{\sec^2{x}}{2u} du\) Now we replace \(\tan x\) and \(\sec^2x\) in terms of \(u\) remembering that \(u^2 = \tan x\) and \(\sec^2{x}=\frac{1}{\cos^2{x}}=1+\tan^2{x}=1+u^4\): \(= \int u \cdot \frac{1 + u^4}{2u} du = \frac{1}{2} \int (1 + u^2) du\)

Evaluate the integral

Now we can evaluate the integral easily: \(\frac{1}{2} \int (1 + u^2) du = \frac{1}{2}(\int 1 du + \int u^2 du)\) \(= \frac{1}{2}(u + \frac{1}{3}u^3)|_{a}^{b}\) Now, let's find the values of \(a\) and \(b\). As \(\int_{0}^{\pi / 4} \sqrt{\tan x} d x\), we have \(x=0\) and \(x=\pi/4\). When \(x=0\), \(u=\sqrt{\tan 0} = 0\). Therefore, \(a=0\). When \(x=\pi/4\), \(u=\sqrt{\tan \frac{\pi}{4}} = 1\). Therefore, \(b=1\). Now substitute the values of \(a\) and \(b\) and evaluate: \(= \frac{1}{2}(1 + \frac{1}{3}(1)^3) - \frac{1}{2}(0 + \frac{1}{3}(0)^3)\) \(= \frac{1}{2} (1 + \frac{1}{3})\) \(= \frac{4}{6}\) \(= \frac{2}{3}\) Thus, the value of the integral \(\int_{0}^{\pi / 4} \sqrt{\tan x} d x\) is \(\frac{2}{3}\).

Key concepts

change of variables

The change of variables is a powerful technique in calculus that helps simplify complex integrals. By transforming the original variable into a new variable, we can convert an integral into a form that is easier to evaluate. In this exercise, we used the substitution \( u = \sqrt{\tan x} \). This new variable \( u \) replaces \( \sqrt{\tan x} \), making the integral simpler.
This process involves:

• Identifying a substitution that makes the integral easier to solve.
• Expressing all parts of the integral, including \( dx \), in terms of the new variable.
• Simplifying the integral using the new variable \( u \).

This technique reduces the complexity of the integration, allowing us to solve it by more straightforward methods.

partial fractions

Once the integral has been transformed to a polynomial in terms of \( u \), partial fraction decomposition may be used. Partial fractions break down a complex rational expression into a sum of simpler fractions, which are easier to integrate. In the step-by-step solution, we used this when we derived \( \frac{1}{2} \int (1 + u^2) \, du \).
Here's how it works:

• Express the polynomial as a sum or difference of simpler fractions.
• Each simpler fraction corresponds to a standard integration form.
• Integrate each of these simple fractions separately.

Partial fractions allow us to take advantage of standard integral forms to find solutions more efficiently.

definite integral

A definite integral allows us to calculate the exact area under a curve between two specified points. For definite integrals, limits of integration are given. In this exercise, those limits were from \( 0 \) to \( \pi/4 \).
Important points:

• The limits of integration \( a \) and \( b \) are substituted for the variable \( x \).
• Once the integral is evaluated, the limits \( a \) and \( b \) are applied to obtain a numerical result.
• The Fundamental Theorem of Calculus provides a link between integration and differentiation for obtaining results.

For our problem, we replaced the limits of \( x \) with their corresponding \( u \) values: \( 0 \) and \( 1 \), allowing us to evaluate the integral and find the area.

substitution method

The substitution method is a technique used to simplify integrals by introducing a new variable. It's particularly useful when dealing with integrals involving products, composites, or trigonometric functions. Through substitution, a complex integral is transformed into a more manageable form.
The procedure includes:

• Choosing an appropriate substitution \( u = f(x) \) to simplify the integral.
• Applying the chain rule to find \( du \) in terms of \( dx \).
• Replacing all occurrences of the original variable with the new variable \( u \).

Substitution not only simplifies the integrals but sometimes turns them into forms solvable by basic integration techniques. In our task, substituting \( u = \sqrt{\tan x} \) streamlined our integration process.