Problem 50 - Calculus Single Variable

Question

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

Step-by-Step Solution

Answer

The integral evaluates to \( x \tan(x) + \ln |\cos(x)| + C \).

Step 1: Identify the Integral Type

The integral given is \( \int x \sec^2(x) \, dx \). This is an integration problem involving a product of a variable \( x \) and a trigonometric function \( \sec^2(x) \). A common technique for these types of integrals is integration by parts.

Step 2: Recall the Integration by Parts Formula

The formula for integration by parts is: \[ \int u \, dv = uv - \int v \, du \]. We need to choose \( u \) and \( dv \) such that the integral becomes simpler to solve.

Step 3: Choose \( u \) and \( dv \)

Let \( u = x \) and \( dv = \sec^2(x) \, dx \). Then, differentiate and integrate respectively: \( du = dx \) and \( v = \tan(x) \) since the integral of \( \sec^2(x) \) is \( \tan(x) \).

Step 4: Apply Integration by Parts Formula

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

Step 5: Solve the Remaining Integral

The remaining integral is \( \int \tan(x) \, dx \). We can solve it by using the identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \), leading to: \[ \int \tan(x) \, dx = \int \frac{\sin(x)}{\cos(x)} \, dx \]. This results in \(-\ln |\cos(x)| + C \), where \( C \) is the constant of integration.

Step 6: Simplify the Expression

Combine the results: \( \int x \sec^2(x) \, dx = x \tan(x) + \ln |\cos(x)| + C \). This is the final solution for the integral.

Key Concepts

Trigonometric Integration

Trigonometric integration is a technique used to solve integrals involving trigonometric functions. These types of integrals often involve sine, cosine, tangent, and their inverses or derivatives. Understanding trigonometric identities and how to manipulate them becomes very important.

• For instance, when dealing with \( \sec^2(x) \), one should know that its integral gives \( \tan(x) \).
• On the other hand, \( \tan(x) \) itself can be rewritten using the identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \), which is crucial for integration.

In our original exercise, the integrand \( x \sec^2(x) \) contains a trigonometric square function, which is why trigonometric integration was specifically useful when breaking it down through integration by parts.

Definite Integrals

Definite integrals calculate the area under a curve for a given interval, denoted \( \int_a^b f(x) \, dx \). While the problem at hand was specifically an indefinite integral task, it's useful to contrast the approach and interpretation with definite integrals.

• Firstly, definite integrals have limits \( a \) and \( b \), which it integrates between.
• As a result, they yield a specific numerical value rather than a general antiderivative plus constant \( C \).

To solve definite integrals using methods like integration by parts or substitution, the process for determining \( u \) and \( dv \) remains unchanged, but you evaluate the resulting expression with respect to the limits \( a \) and \( b \). This evaluation will yield the exact area under the curve within this interval.

Indefinite Integrals

Indefinite integrals are a core concept in calculus used to find the most general antiderivative of a function. These integrals do not have defined limits, thus are expressed with a "plus C" to denote any constant.The integral in the original exercise \( \int x \sec^2(x) \, dx \) is indefinite. The goal was to determine a general formula that represents the family of antiderivatives. Here’s a key takeaway:

• Indefinite integrals can always be expressed as \( F(x) + C \), where \( F(x) \) is the antiderivative.
• Using integration by parts is essential when straightforward antiderivatives are difficult to determine, especially when products of functions like \( x \sec^2(x) \) are involved.

Through the steps of applying the integration by parts method, identifying \( u \), \( dv \), and integrating, we ended up with the expression: \( x \tan(x) + \ln |\cos(x)| + C \). This result highlights the nature of indefinite integrals as a representation of a range of possible functions.