Problem 9 - Thomas Calculus

Question

Evaluate the integrals. \(\int x \sec ^{2} x d x\)

Step-by-Step Solution

Answer

\( \int x \sec^2 x \, dx = x \tan x + \ln |\cos x| + C \)

Step 1: Identify the Integration Technique

The integral \( \int x \sec^2 x \, dx \) requires integration by parts. Integration by parts is a method that involves selecting parts of the integral as \( u \, \) and \( dv \) and using the formula \( \int u \, dv = uv - \int v \, du \).

Step 2: Choose Parts for Integration by Parts

Select \( u = x \) and therefore \( du = dx \); select \( dv = \sec^2 x \, dx \), which means \( v = \tan x \) as the derivative of \( \tan x \) is \( \sec^2 x \).

Step 3: Apply the Integration by Parts Formula

Substitute the chosen \( u, du, v, \) and \( dv \) into the integration by parts formula: \[ \int x \sec^2 x \, dx = x \tan x - \int \tan x \, dx. \]

Step 4: Evaluate the Remaining Integral

The remaining integral \( \int \tan x \, dx \) can be found by using the identity \( \int \tan x \, dx = -\ln |\cos x| + C \), hence, \[ \int \tan x \, dx = -\ln |\cos x| + C. \]

Step 5: Finalize the Solution

Substitute the result from Step 4 into the integration by parts formula, leading to: \[ \int x \sec^2 x \, dx = x \tan x + \ln |\cos x| + C. \]

Key Concepts

Integral Calculus

Integral calculus is a core part of calculus focused on the concept of anti-differentiation. It revolves around finding a function, called the antiderivative or integral, to express the accumulation of quantities. In simpler terms, it is used to find areas under curves, solve differential equations, and compute total quantities when given rates of change.
Integral calculus is essentially about reversing the process of differentiation. Instead of finding how rapidly a function changes, you are finding the original function itself. It's the undoing of the derivative to find the integral, which can represent

• area under a curve
• a displacement from velocity
• total growth from a rate

Using techniques like integration by parts, substitution, and partial fractions helps tackle more complex integrals that can't be solved directly. By understanding integral calculus, one can analyze how different quantities accumulate over time or space.

Trigonometric Integration

Trigonometric integration specifically deals with integrals involving trigonometric functions. These integrals are common in calculus due to the periodic nature of trigonometric functions, which appear in many real-world applications, from physics to engineering.

In the exercise given, the integral \( \int x \sec^2 x \; dx \)involves \(\sec^2 x\), a trigonometric function with useful properties. Here’s a basic look at how trigonometric functions are integrated:

• The integral of \(\sec^2 x\) is known directly: \(\tan x\), as its derivative is \(\sec^2 x\).
• Other identities, like \( \sin^2 x + \cos^2 x = 1 \), help simplify behavioral functions within integrals.

Understanding these integrals is crucial when solving problems involving waves, oscillations, or any scenario where periodicity is key. Mastery of trigonometric integration enables the solution of complex calculus problems, helping you evaluate trigonometric functions' behaviors and interactions.

Integration Techniques

Integration techniques are the different methods one uses to compute the integral of a function which cannot be integrated directly, or when the function appears too complex. The technique of Integration by Parts is particularly important, especially when dealing with the product of functions, as in the given integral \( \int x \sec^2 x \, dx \).

Here's a brief overview of the technique:

• Integration by Parts is based on the formula \(\int u \, dv = uv - \int v \, du\).
• You choose \( u \) and \( dv \) from the original integral such that their differentiation and integration are straightforward.
• In the sample problem, \( u = x \) and \( dv = \sec^2 x \, dx \); this choice simplifies the problem, leading to more manageable components.

Advanced integration techniques like Integration by Parts allow for critical problem-solving capabilities in areas like physics and engineering. Applying these techniques strategically eases learning complex integrations and achieving precise results.