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Chapter 7: Problem 13

Find the following indefinite and definite integrals. \(\int_{0}^{\sqrt{\pi} / 2} x \sec ^{2}\left(x^{2}\right) \tan \left(x^{2}\right) d x\)

Short Answer

Expert verified
The definite integral of the given function is \( \frac{1}{2} \).

Step by step solution

01

Identify the Integration Technique

The integral is of the form \( \int x \sec^2(x^2) \tan(x^2) \, dx \), which suggests the use of substitution due to the \( \sec^2(u) \tan(u) \) expression.
02

Determine Substitute Variables

Let \( u = x^2 \), then \( \frac{du}{dx} = 2x \) or \( du = 2x \, dx \). Thus, \( x \, dx = \frac{1}{2} \, du \).
03

Rewrite the Integral with Substitution

Substitute \( u = x^2 \) and \( x \, dx = \frac{1}{2} \, du \), so the integral becomes \( \frac{1}{2} \int \sec^2(u) \tan(u) \, du \).
04

Evaluate the Indefinite Integral

The integral \( \int \sec^2(u) \tan(u) \, du \) is \( \tan^2(u)/2 + C \), considering the basic identity used in integration \( \int \sec^2(x) \, dx = \tan(x) + C \). Thus, the indefinite integral is \( \frac{1}{2} \tan^2(u) + C \).
05

Back Substitute to Original Variable

Since \( u = x^2 \), substitute back to get \( \frac{1}{2} \tan^2(x^2) + C \).
06

Evaluate Definite Integrals with Limits

Now, evaluate the definite integral \( \int_{0}^{\sqrt{\pi}/2} x \sec^2(x^2) \tan(x^2) \, dx \). In terms of \( u \), the limits transform from \( x = 0 \) to \( u = 0 \) and from \( x = \sqrt{\pi}/2 \) to \( u = \pi/4 \).
07

Calculate Definite Integral

Substitute the limits into \( \frac{1}{2} \tan^2(u) \), resulting in \[ \frac{1}{2} [ \tan^2(\pi/4) - \tan^2(0) ] = \frac{1}{2} (1 - 0) = \frac{1}{2}. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Indefinite Integration
Indefinite integration is the process of finding an antiderivative of a function. Unlike definite integration, it does not involve specific limits of integration. The result of indefinite integration includes a constant of integration, represented by the symbol \( C \). This constant accounts for any constant value that might have differentiated to zero in the original function. Here鈥檚 how it plays out:
  • Given a function \( f(x) \), the indefinite integral is represented by \( \int f(x) \, dx \).
  • The goal is to find a function \( F(x) \) such that \( F'(x) = f(x) \).
  • The general solution is then \( F(x) + C \), where \( C \) is any constant value.
Indefinite integrals are one of the building blocks of calculus, used extensively in both theoretical and applied mathematics to determine a family of functions that differentiate back to the original function.
Definite Integration
Definite integration involves finding the integral of a function over a specific interval. It is a calculation of the net area under a curve across a given range. Unlike indefinite integration, definite integration results in a specific numerical value. The integral is calculated using limits of integration:
  • The notation \( \int_{a}^{b} f(x) \, dx \) represents the definite integral of \( f(x) \) from \( x = a \) to \( x = b \).
  • This calculates the net area between the function \( f(x) \) and the x-axis, from the starting point \( a \) to the endpoint \( b \).
  • The result is a real number that represents this total area.
Definite integration is essential in physics and engineering where it is used to calculate quantities like area, volume, and other accumulative quantities that are derived from rates of change.
Integration Techniques
Integration techniques are methods employed to solve integrals that are not straightforward. Since there isn't a universal method to integrate every function, mathematicians have devised various techniques suitable for different forms of integrals. These are some common techniques:
  • Substitution Method: Used when an integral contains a composite function. It involves substituting parts of the integral to simplify the integral into an easily calculable form.
  • Integration by Parts: Utilized when the integrand is a product of two functions. It is based on the product rule for differentiation and can simplify many complex integrals.
  • Partial Fraction Decomposition: Useful for rational functions, this technique involves breaking down a complex fraction into simpler parts that are easier to integrate.
  • Trigonometric Substitution: Applies to certain integrals involving square roots, where trigonometric identities simplify the integral.
These techniques are crucial for solving complex integrals, each method having its own specific applications and forms of usage depending on the type of function being integrated.
Substitution Method
The substitution method is a powerful, often-used technique in integration, especially when dealing with composite functions. It simplifies an integral by replacing a complex expression with a single variable. Here is how it works:
  • Select a Substitution Variable: Identify an inner function within the integrand and assign it a new variable, often new variable 'u'. For the given integral, \( u = x^2 \).
  • Differentiate and Solve for dx: Compute \( du \) in terms of \( dx \). If \( u = x^2 \), then \( du = 2x \, dx \), so \( x \, dx = \frac{1}{2} \, du \).
  • Rewrite the Integral: Substitute \( u \) and \( dx \) in the original integral to get a new integral in terms of \( u \) which is easier to handle.
  • Integrate: Once the integral is in its simpler form, perform the integration. For example, \( \int \sec^2(u) \tan(u) \, du \) becomes \( \frac{1}{2} \tan^2(u) + C \).
  • Back-Substitute: Convert back to the original variable once the integration is done, replacing \( u \) with \( x^2 \).
The substitution method is especially effective when dealing with integrals involving functions that are derivatives of one another, like \( \sec^2(u) \) and \( \tan(u) \). It significantly simplifies the integration process, making it an invaluable tool in calculus.

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