/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 53 Evaluate the integrals in Exerci... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 53

Evaluate the integrals in Exercises \(51-56\) by making a substitution (possibly trigonometric) and then applying a reduction formula. $$ \int_{0}^{1} 2 \sqrt{x^{2}+1} d x $$

Short Answer

Expert verified
The integral evaluates to \( \ln(1 + \sqrt{2}) + \sqrt{2} \).

Step by step solution

01

Identify a Substitution

For the integral \( \int 2 \sqrt{x^{2}+1} \, dx \), we can make the substitution \( x = \sinh(t) \) because \( \sqrt{x^2 + 1} = \sqrt{\sinh^2(t) + 1} = \cosh(t) \). Therefore, \( dx = \cosh(t) \, dt \).
02

Change the Limits of Integration

When \( x = 0 \), \( \sinh(t) = 0 \) implies \( t = 0 \). When \( x = 1 \), \( \sinh(t) = 1 \) implies \( t = \sinh^{-1}(1) \). Thus, the limits change from \( x = 0 \) to \( x = 1 \) to \( t = 0 \) to \( t = \sinh^{-1}(1) \).
03

Substitute and Simplify the Integral

Substitute \( x = \sinh(t) \) into the integral: \( \int 2 \sqrt{\sinh^2(t) + 1} \cosh(t) \, dt = \int 2 \cosh^2(t) \, dt \).
04

Apply the Hyperbolic Identity

We know that \( \cosh^2(t) = \frac{1+\cosh(2t)}{2} \). Thus, the integral becomes \( \int 2 \left(\frac{1+\cosh(2t)}{2}\right) \ dt = \int (1 + \cosh(2t)) \, dt \).
05

Integrate Using Basic Formulas

Integrate each term: \( \int 1 \, dt = t \) and \( \int \cosh(2t) \, dt = \frac{1}{2} \sinh(2t) \). Therefore, the integral evaluates to \( t + \frac{1}{2} \sinh(2t) \).
06

Back-Substitute and Evaluate

Substitute back the limits, \( t = 0 \) and \( t = \sinh^{-1}(1) \), into \( t + \frac{1}{2} \sinh(2t) \). Calculate to find the final value: \[ \left[ \sinh^{-1}(1) + \frac{1}{2} \sinh(2(\sinh^{-1}(1))) \right] - \left[ 0 + \frac{1}{2} \sinh(2 \, \times \, 0) \right] \].
07

Final Calculation

\( \sinh^{-1}(1) = \ln(1 + \sqrt{2}) \) and \( \sinh(2 \sinh^{-1}(1)) = 2 \sinh(\sinh^{-1}(1)) \cosh(\sinh^{-1}(1)) = 2 \times 1 \times \sqrt{2} = 2\sqrt{2} \). So the expression becomes \[ \ln(1 + \sqrt{2}) + \sqrt{2} \].

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Key Concepts

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

Hyperbolic Functions
Hyperbolic functions are analogs of trigonometric functions but for a hyperbola, rather than a circle. They are useful in calculus.
  • Common hyperbolic functions include \( \sinh(t) = \frac{e^t - e^{-t}}{2} \) and \( \cosh(t) = \frac{e^t + e^{-t}}{2} \).
  • Key identity: \( \cosh^2(t) - \sinh^2(t) = 1 \), similar to the Pythagorean identity in trigonometry.
  • In some integrals, substituting a variable with a hyperbolic function can simplify the integration process. For instance, if \( x = \sinh(t) \, \ dx = \cosh(t) \, dt \), then it converts functions involving \( \sqrt{x^2 + 1} \) efficiently.
These simplifications can transform a complex problem into a more manageable one, such as converting \( \sqrt{x^2 + 1} \) into \( \cosh(t) \) through hyperbolic substitution.
Reduction Formula
Reduction formulas are powerful tools in calculus for breaking down complicated integrals into simpler parts. They are iterative relations that express an integral \( I_n \) in terms of a similar integral \( I_{n-1} \).
  • Typically used in solving integrals involving powers of functions.
  • Commonly applied in sequences where each step reduces the problem by one or more degrees.
  • For instance, a reduction formula may express an integral of the nth power in terms of the \( n-1 \) power, thus creating a solvable pattern.
By applying these formulas, integrals become approachable by working through repeatedly reduced forms until they reach an easily integrable function.
Trigonometric Substitution
Trigonometric substitution is a technique where we replace a variable with a trigonometric function to simplify integrals, often involving radicals.
  • Used primarily for integrals of forms involving \( \sqrt{a^2 - x^2}, \sqrt{a^2 + x^2} \, \sqrt{x^2 - a^2} \).
  • The substitution aims to convert complex expressions into trigonometric ones to take advantage of trigonometric identities.
  • For example, for an integral with \( \sqrt{x^2 + 1} \, x = \tan(\theta) \) or \( x = \sinh(t) \) reduces it to trigonometric or hyperbolic forms.
This method turns the difficult task of integrating complex radicals into simpler trigonometric or hyperbolic equations that are easier to solve.
Definite Integrals
Definite integrals involve calculating the accumulation of quantities, specifically finding the net area between the function and the x-axis over a defined interval.
  • They are usually represented as \( \int_{a}^{b} f(x) \, dx \, \) where \( a \) and \( b \) are the limits of integration.
  • They provide exact values for total accumulation, such as total distance using a velocity function over time.
  • In the substitution method, limits need to change according to the new variable of substitution.
In our exercise, the limits changed from \( x = 0 \) to \( t = 0 \, \ x = 1 \) to \( t = \sinh^{-1}(1) \, \) a crucial step in solving the problem.

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