Problem 435 - Calculus Volume 1

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 \(\csc ^{2} x .\) )

Step-by-Step Solution

Answer

\(-\frac{\sqrt{2}}{2} \tan^{-1}(\cot x) + C\).

Step 1: Identify the Transformation

We are given the substitution \(u=\sqrt{2} \cot x\). Recall that the trigonometric identity \(1 + \cot^2 x = \csc^2 x\) will be instrumental in simplifying the integral.

Step 2: Multiply Integrand by \(\csc^2 x\)

To transform the integral \(\int \frac{dx}{1+\cos^2 x} \), multiply the numerator and the denominator by \(\csc^2 x\). This gives us \(\int \frac{\csc^2 x \, dx}{\csc^2 x + \cot^2 x}\).

Step 3: Simplify Using Identity

Use the identity \(\csc^2 x = 1 + \cot^2 x\). The integral transforms to \(\int \frac{\csc^2 x \, dx}{1 + 2\cot^2 x}\) as \(1 + \cos^2 x = \frac{1 + \cot^2 x}{\csc^2 x}\).

Step 4: Express in Terms of \(u\)

Substitute \(u = \sqrt{2} \cot x\), hence \(\cot x = \frac{u}{\sqrt{2}}\). Therefore, \(\csc^2 x dx = -\frac{\sqrt{2}}{u^2+2} du\).

Step 5: Perform Integration

Replace \(dx\) in the integral to get \(-\int \frac{\sqrt{2}}{u^2 + 2} \, du\). Solving this gives \(-\frac{\sqrt{2}}{2} \tan^{-1}(\frac{u}{\sqrt{2}}) + C\), where \(C\) is the constant of integration.

Step 6: Re-substitute for Original Variable

Since \(u = \sqrt{2} \cot x\), substitute back to express the answer in terms of \(x\). Thus, \(-\frac{\sqrt{2}}{2} \tan^{-1}(\cot x) + C\).

Key Concepts

Integral Calculus

Integral calculus focuses on the concept of integration, a fundamental aspect of calculus. Integration, along with differentiation, forms the basis of many calculations in mathematics and physics. It allows us to find areas under curves, volumes, central points, and numerous other things. In essence, when you think about integration, imagine accumulating or summing very small quantities to figure out a whole.
Its applications are extensive:

• Calculating areas and volumes.
• Determining averages and center points.
• Data smoothing techniques in statistics.
• Modeling in physics for motion, growth, or decay processes.

Understanding integral calculus is crucial for solving the given integral problem. We start by transforming the problem into a more manageable form using substitution, which then enables us to apply known calculus methods to evaluate it efficiently.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable. These identities play a vital role in simplifying integrals, particularly those involving trigonometric expressions.
Some of the essential identities you'll encounter often include:

• \( \sin^2 x + \cos^2 x = 1 \)
• \( 1 + \tan^2 x = \sec^2 x \)
• \( 1 + \cot^2 x = \csc^2 x \)

In the provided exercise, the identity \(1 + \cot^2 x = \csc^2 x\) helps simplify our integral by converting a cotangent expression into a cosecant one. This simplification is vital for transforming the integral into a solvable form using known techniques. The identity helps us manage trigonometric terms and manipulate the function so the integration is feasible.

Integration Techniques

Integration techniques are strategies for solving integrals, and they include methods like substitution, integration by parts, and partial fractions. These techniques are necessary when standard elementary methods do not directly apply.
In this exercise, **trigonometric substitution** is the primary technique. By using the substitution \( u = \sqrt{2} \cot x \), the integral becomes more manageable. This method involves changing variables to transform the given integral into one that is simpler to evaluate.
Additionally:

• Recognize when substitution will simplify the integral.
• Apply the substitution to transform the integral entirely.
• After solving, re-substitute to return to the original variable.

This approach reduces complexity and leverages trigonometric identities to handle intricate trigonometric expressions. By reorganizing these expressions and integrating step-by-step, we arrive at a solution that might otherwise be tricky to find directly.