Question

Use the substitution \(u=\sqrt{2} \cot x\) and the identity \(1+\cot ^{2} x=\csc ^{2} x\) to evaluate \(\int \frac{d x}{1+\cos ^{2} x}\). (Hint: Multiply the top and bottom of the integrand by \(\left.\csc ^{2} x .\right)\)

Short answer

The integral evaluates to an expression involving arctangent or logarithm after substitutions.

Step-by-step solution

Rewrite the Integrand

We begin with the integral \( \int \frac{dx}{1+\cos^2 x} \). To simplify, we multiply both the numerator and the denominator by \( \csc^2 x \). Thus, the integral becomes \( \int \frac{\csc^2 x \, dx}{\csc^2 x + \cos^2 x \cdot \csc^2 x} \).

Use Identity

Recall the identity \( 1 + \cot^2 x = \csc^2 x \). Substituting this identity into our integral\( \int \frac{\csc^2 x \, dx}{\csc^2 x + (1 - \sin^2 x)} = \int \frac{\csc^2 x \, dx}{2 - \sin^2 x} \).

Substitute Variable

Let's employ the substitution \( u = \sqrt{2} \cot x \). Then, \( \frac{du}{dx} = -\sqrt{2} \csc^2 x \). Thus, \( dx = -\frac{du}{\sqrt{2} \csc^2 x} \). Substitute this into the integrand:\( \int \frac{\csc^2 x (-du)}{\sqrt{2}(2 - \cot^2 x)} \).

Simplify the Integral

Notice that since \( u = \sqrt{2} \cot x \), we have \( \cot^2 x = \frac{u^2}{2} \). The integral now becomes: \( \int \frac{-du}{\sqrt{2} (2 - \frac{u^2}{2})} = \int \frac{-du}{\sqrt{2} (2 - \frac{u^2}{2})} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \int \frac{-\sqrt{2} \, du}{4 - u^2} \).

Integrate Using Partial Fraction Decomposition

Recognize that \( 4-u^2 \) can be factored into \((2-u)(2+u)\). Using partial fraction decomposition, we express \( \frac{-\sqrt{2}}{4-u^2} \) as a sum of simpler fractions and integrate each individually. This will typically involve completing the square or looking up integral formulas.

Evaluate the Result

After integrating, substitute back \( \cot x = \frac{u}{\sqrt{2}} \) to express the solution in terms of \( x \). Any constants of integration should be adjusted accordingly. The resulting expression will be in terms of \( x \).

Key concepts

Substitution Method

The substitution method is one of the fundamental techniques for solving integrals. It involves changing variables to transform a given integral into a simpler form. In this problem, we used the substitution \( u = \sqrt{2} \cot x \). This approach helps to simplify the integral by expressing it with respect to a new variable \( u \), making integration more manageable.

To perform substitution effectively, follow these steps:

• Identify a part of the integrand that can be replaced by \( u \). In this case, \( \cot x \) is chosen due to the similarity to the identity \( 1 + \cot^2 x = \csc^2 x \).
• Express \( dx \) in terms of \( du \) by differentiating \( u = \sqrt{2} \cot x \) to get \( \frac{du}{dx} = -\sqrt{2} \csc^2 x \). Thus, \( dx = -\frac{du}{\sqrt{2} \csc^2 x} \).
• Substitute \( u \) and the expression for \( dx \) back into the integral, rewriting it in terms of \( u \).

This method simplifies the integration process, eventually aiding the solution of complex functions.

Trigonometric Identities

Trigonometric identities are essential tools in calculus, particularly useful in solving integrals involving trigonometric functions. In this exercise, we utilized the identity \( 1 + \cot^2 x = \csc^2 x \) to simplify the integrand.

Here's how these identities assist in integration:

• They allow you to replace trigonometric expressions with equivalent forms that are easier to integrate or simplify.
• In this problem, after substituting \( \cot x \), we used \( \cot^2 x = \frac{u^2}{2} \) as a result of the substitution \( u = \sqrt{2} \cot x \).
• These transformations reduce the complexity of integrals with multiple trigonometric components.

Understanding and applying these identities can significantly expand your ability to solve integrals involving trigonometric functions.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a complicated rational expression as a sum of simpler fractions, making integration more straightforward. Here, it helped transform \( \frac{-\sqrt{2}}{4-u^2} \) into fractions that are easier to integrate.

Using partial fraction decomposition involves these steps:

• Factor the denominator if possible. In our case, \( 4-u^2 \) factors into \((2-u)(2+u)\).
• Express the fraction as a sum of simpler fractions: \( \frac{A}{2-u} + \frac{B}{2+u} \).
• Determine the values of \( A \) and \( B \) by solving equations derived by substituting various values of \( u \) or by matching coefficients.
• Integrate each of these simpler fractions individually.

This transformation simplifies the integration process, allowing you to handle complex rational expressions systematically.