Problem 40 - Essential Calculus

Question

Find the indefinite integral. $$ \int \sec (2-x) \tan (2-x) d x $$

Step-by-Step Solution

Answer

The result of the indefinite integral is \( \sec (2-x) + C \).

Step 1: Recognize the pattern

Notice that the integral is in the form of \( \int \sec u \tan u \, du \) , where \(u = 2-x\). This is a standard integral and it directly integrates to \( \sec u + C\), where C is the constant of integration.

Step 2: Perform the integration

Now we substitute u=2-x back into our integral. The result is \( \sec (2-x) + C \).

Step 3: Final Result

The result of the integral is \( \sec (2-x) + C \). Since the original problem doesn't provide any limits of integration or more specific context, the solution can't be simplified much further than this.

Key Concepts

Integral Calculus

Integral calculus is one of the two primary branches of calculus, with the other being differential calculus. The focus of integral calculus is on finding the accumulative sum of areas or quantities and solving problems related to accumulation. In essence, integration is the inverse operation of differentiation; where differentiation gives us the rate at which a quantity changes, integration provides the total change given the rate.

When we integrate a function, we're finding the indefinite integral or antiderivative, which represents a family of functions whose derivative is the original function. An indefinite integral is denoted as , and it comes with a '+ C', the constant of integration, symbolizing that an infinite number of antiderivatives exist, each differing by a constant value.

Integration Techniques

Various techniques in integration allow us to solve integrals that are not immediately recognizable or directly integrable. Some common techniques include substitution, integration by parts, partial fraction decomposition, and trigonometric integration. Each technique has its own set of rules and applications, making them more suitable for certain types of integrals over others.

Substitution, for instance, simplifies the integral by transforming it into a new variable, as seen in the provided exercise. Here, we substituted the variable 'u' for '2-x', allowing us to recognize the integral as a standard form. The ability to spot patterns and apply the appropriate integration technique is a critical skill in calculus that becomes easier with practice and familiarity with a variety of integral forms.

Trigonometric Integrals

Trigonometric integrals involve integrating functions that contain trigonometric functions such as , and . These integrals often appear complex, but they can be simplified using trigonometric identities and substitution.

In the exercise, we demonstrate how to integrate a function that includes and , where our choice of substitution simplified the expression into a basic integral form that could be solved easily. Recognizing that the integrals of and can be directly integrated to and , respectively, is essential in solving trigonometric integrals. Advanced trigonometric integrals may require further techniques like trigonometric substitution or breaking the function into partial fractions.