Problem 98 - Calculus Early Transcendentals

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 \(\int_{0}^{\pi / 4} \sqrt{\tan x} d x\)

Step-by-Step Solution

Answer

Question: Evaluate the definite integral of the given expression from \(0\) to \(\pi/4\): \(\int \sqrt{\tan x} dx\) Answer: \(\frac{3}{5}\)

Step 1: Perform the change of variables

Let \(u = \sqrt{\tan x}\). We need to find a substitution for \(dx\) in terms of \(du\). To do this, first express \(\tan x\) in terms of \(u\). We have, $$ \tan x = u^2 $$ Now, differentiate both sides with respect to \(x\): $$ \frac{d}{d x}(\tan x) = \frac{d}{d x}(u^2) $$ Using chain rule and remembering \(\sec^2 x = \frac{d}{d x}(\tan x)\), $$ \sec^2 x dx = 2u \, du $$

Step 2: Replace \(dx\) in terms of \(du\) in the integral

Now let's replace \(dx\) in the integral: $$ \int \sqrt{\tan x} dx = \int u \cdot \frac{\sec^2 x dx}{2u} $$ Cancelling \(u\) and factor 2: $$ \frac{1}{2}\int \sec^2 x \, du $$

Step 3: Express \(\sec^2 x\) in terms of \(u\)

Recall that \(\sec^2 x = 1 + \tan^2 x\). We can rewrite this in terms of \(u\) as: $$ \sec^2 x = 1 + u^4 $$ Now, substitute this into the integral: $$ \frac{1}{2} \int (1 + u^4) du $$

Step 4: Evaluate the integral

Integrating with respect to \(u\) gives: $$ \frac{1}{2} \int (1 + u^4) du = \frac{1}{2} (u + \frac{1}{5} u^5) + C = \frac{1}{2}(u + \frac{1}{5}u^5) + C $$ Now, substitute back \(u = \sqrt{\tan x}\) to get the result in terms of \(x\): $$ \frac{1}{2}(\sqrt{\tan x} + \frac{1}{5} (\tan x)^{\frac{5}{2}}) + C $$

Step 5: Evaluate the definite integral

Finally, evaluate the definite integral from \(0\) to \(\pi/4\): $$ \int_{0}^{\pi / 4} \sqrt{\tan x} dx = \frac{1}{2}(\sqrt{\tan x} + \frac{1}{5} (\tan x)^{\frac{5}{2}}) \Big|_0^{\pi/4} $$ $$ =\frac{1}{2} [(\sqrt{\tan {\pi/4}} + \frac{1}{5} (\tan {\pi/4})^{\frac{5}{2}}) - (\sqrt{\tan 0} + \frac{1}{5} (\tan 0)^{\frac{5}{2}})] $$ As we know, \(\tan 0 = 0\) and \(\tan (\pi/4) = 1\). Therefore, $$ \int_{0}^{\pi / 4} \sqrt{\tan x} dx = \frac{1}{2} [(\sqrt{1} + \frac{1}{5} (1)^{\frac{5}{2}}) - (\sqrt{0} + \frac{1}{5} (0)^{\frac{5}{2}})] = \frac{1}{2}(1+\frac{1}{5}) = \frac{3}{5} $$ Thus, the value of the given integral from \(0\) to \(\pi/4\) is \(\frac{3}{5}\).

Key Concepts

Change of Variables

The change of variables is a technique used in calculus to simplify integrals by substituting a part of the integrand (the expression inside the integral) with a new variable. This substitution transforms the integral into a form that's easier to solve. In this exercise, we use the substitution

• Let: \( u = \sqrt{\tan x} \)

This substitution implies that \( \tan x \) can be expressed as \( u^2 \). By differentiating both sides, we find that

• \( \sec^2 x \, dx = 2u \, du \)

The choice of substitution is often guided by the aim of simplifying the expression considerably. The key is to find a substitution where the new integral is easier to evaluate. Here, the change in variables transforms a complex trigonometric function into a simple polynomial expression imposed on integrals. This method is instrumental when dealing with integrals involving composite functions.

Partial Fractions

Partial fractions involve decomposing a complex rational function into a sum of simpler rational expressions. This method greatly aids in integrating functions that are difficult to handle. After changing variables in the original problem to \( u = \sqrt{\tan x} \), we arrive at an integral that lends itself well to partial fraction decomposition:

• \( \frac{1}{2} \int (1 + u^4) \, du \)

Instead of tackling the entire complex fraction, the strategy here is to first break it down. Unfortunately, here it doesn't decompose further, but this is because the polynomial is already a straightforward form for integration. Essentials of this concept lie in the ability to recognize when a polynomial fraction can be expanded into simpler, more easily integrable parts. Overall, this approach makes calculus much more approachable, particularly in dealing with more involved expressions.

Definite Integrals

Definite integrals are a key tool used to calculate the accumulation of quantities, represented as the area under a curve between two specific points. In this exercise, we explore the definite integral

• \( \int_{0}^{\pi / 4} \sqrt{\tan x} \, dx \)

After performing the change of variables and integrating the new expression, we substitute back to the original variable and evaluate at the bounds, resulting in:

• \( \frac{3}{5} \)

The process involves substituting \( u = \sqrt{\tan x} \) and converting our bounds accordingly. Evaluating definite integrals often involves key calculus operations: substitution, differentiation, and simplification. This integral calculation understanding is crucial because it allows us to find exact values for areas or accumulations, providing precise results which are essential in different applied sciences and engineering tasks.