Properties Of Integrals & Techniques

Unit: Integration & Accumulation of Change

Chapter: Properties of Integrals & Techniques

Reference: – Riemann sums, Definite integrals, Fundamental theorem, Antiderivatives, Area under a curve, Accumulation functions, Average value of functions, Mean value theorem for Integrals, Properties & Estimation, Trapezoidal rule, Simpson's Rule, Application & Motion problems.

After studying this chapter, you should be able to:

  • Linearity of Integrals & Constant multiple rules.
  • Bounds, Limits & Reversing the Bounds.
  • Integrals of Symmetric & Odd Functions.
  • Integrals of Periodic Functions & Absolute Values.

 

Linearity of Integration & Constant Multiple Rules

  1. Linearity of Integrals:
    • The linearity property states that the integral of a sum or difference of functions is equal to the sum or difference of their integrals.
    • Mathematically, if f(x) and g(x) are integrable functions, and a and b are constants, then: ∫[af(x) + bg(x)] dx = a∫f(x) dx + b∫g(x) dx.
    • In simpler terms, you can integrate each function separately and then combine the results using addition or subtraction.
  2. Constant Multiple Rule:
    • The constant multiple rules state that the integral of a constant time of a function is equal to the constant times of the integral of the function.
    • Mathematically, if f(x) is an integrable function and c is a constant, then: ∫cf(x) dx = c∫f(x) dx.
    • This rule allows you to bring constants outside of the integral sign.

Bounds, Limits & Reversing the Bounds:

  1. Bounds of Integration:
    • When evaluating definite integrals, the bounds of integration specify the interval over which the integration is performed.
    • The lower bound represents the starting point of the interval, and the upper bound represents the endpoint.
    • For example, in the definite integral ∫[a, b] f(x) dx, 'a' is the lower bound, and 'b' is the upper bound.
  2. Limits of Integration:
    • Integrals can be evaluated over finite intervals or unbounded intervals.
    • Finite intervals have definite limits of integration, where the lower and upper bounds are specific numbers.
    • Unbounded intervals can extend to infinity (∞) or negative infinity (-∞), denoted by ±∞.
    • For example, ∫[0, ∞] f(x) dx represents integration from 0 to infinity.
  3. Reversing the Bounds:
    • Reversing the bounds in an integral change the sign of the result.
    • Mathematically, if a and b are the original bounds of integration, then: ∫[b, a] f(x) dx = -∫[a, b] f(x) dx.
    • This property arises from the fact that reversing the bounds changes the orientation of the integration interval, leading to a negative sign in the result.

 

Integrals of Symmetric & Odd functions: –

 

  1. Integrals of Symmetric Functions:
    • A function is symmetric about the y-axis if it has the property that for every value of x in its domain, the corresponding y-values on opposite sides of the y-axis are equal.
    • When integrating a symmetric function over an interval symmetric about the y-axis, the result is zero.
    • Mathematically, if f(x) is a symmetric function and the interval of integration is [-a, a], then: ∫[-a, a] f(x) dx = 0.
    • This property arises because the positive and negative areas on either side of the y-axis cancel each other out.
  2. Integrals of Odd Functions:
    • An odd function is symmetric about the origin (0, 0), meaning that for every value of x in its domain, f(x) = -f(-x).
    • When integrating an odd function over an interval symmetric about the origin, the result is also zero.
    • Mathematically, if f(x) is an odd function and the interval of integration is [-a, a], then: ∫[-a, a] f(x) dx = 0.
    • This property arises because the positive and negative areas on either side of the origin cancel each other out.

 

Integration of Rational Function & Improper Integrals:

 

  1. Integration of Rational Functions:
    • A rational function is a ratio of two polynomial functions. The process of integrating a rational function involves finding an antiderivative or indefinite integral of the function.
    • To integrate a rational function, various techniques are used, including polynomial long division, partial fractions decomposition, and substitution.
    • Polynomial long division is used to express the rational function as a sum of a polynomial and a proper fraction, which can be integrated separately.
    • Partial fractions decomposition is a method used to decompose a rational function into simpler fractions, making it easier to integrate.
    • After decomposition, the resulting fractions can be integrated using basic integration rules.
    • Substitution is often employed to simplify the integral by replacing a variable with a new variable or expression.
  2. Improper Integrals:
    • An improper integral is an integral with infinite limits or an integrand that has discontinuities within the interval of integration.
    • Improper integrals are evaluated by taking limits as one or both of the bounds approach infinity or a point of discontinuity.
    • There are two types of improper integrals: a. Type 1: Infinite Intervals: In these integrals, one or both of the bounds of integration are ±∞. b. Type 2: Discontinuous Integrand: In these integrals, the integrand has a discontinuity within the interval of integration.
    • To evaluate improper integrals, techniques such as limit evaluation, breaking the integral into several parts, or using comparison tests are applied.
    • The limits involved in improper integrals ensure that the integral converges to a finite value or diverges to infinity.

Integration of Periodic Function & Absolute values

  1. Integration of Periodic Functions:
    • A periodic function is a function that repeats itself after a certain interval called a period.
    • When integrating a periodic function over one period, the result is equal to the integral over any other period.
    • Mathematically, if f(x) is a periodic function with period P, then for any real numbers a and b: ∫[a, a + P] f(x) dx = ∫[b, b + P] f(x) dx.
    • This property arises from the fact that the positive and negative areas of the periodic function cancel each other out over one period.
    • It allows us to evaluate the integral of a periodic function by considering a single period, simplifying the calculation.
  2. Integration of Absolute Values:
    • When integrating a function involving absolute values, the integral can be related to the original function by splitting the integral at the points where the function changes sign.
    • Mathematically, if f(x) is a function, then: ∫[a, b] |f(x)| dx = ∫[a, c] f(x) dx + ∫[c, b] -f(x) dx, where c is a point in the interval [a, b] where f(x) changes sign.
    • This property arises because the absolute value function removes the sign of the original function, and integrating the negative part cancels out the negative areas of the function.
    • It allows us to simplify the integration of functions involving absolute values by considering the different intervals where the function is positive or negative separately.

Example: – Using the Linearity of Integrals and Constant Multiple Rule

 

Find the integral of the function F(x) = 3x2 + 2sin(x) – 4cos(x) over the interval [0, π].

Solution:

To find the integral of F(x) over the given interval, we can apply the linearity of integrals by integrating each term separately and then summing them.

 

∫[0, π] F(x) dx = ∫[0, π] (3x2) dx + ∫[0, π] (2sin(x)) dx – ∫[0, π] (4cos(x)) dx

Using the power rule of integration, the first term becomes:

= [x3] from 0 to π

= π3 – 0

= π3

The second term involves integrating sin(x), which is a known function:

= [-2cos(x)] from 0 to π

= -2cos(π) + 2cos(0)

= -2(-1) + 2(1)

= 4

Similarly, for the third term involving cos(x), we have:

= [-4sin(x)] from 0 to π

= -4sin(π) + 4sin(0)

= -4(0) + 4(0)

= 0

Summing up the individual integrals, we get:

∫[0, π] F(x) dx = π3 + 4 + 0

= π3 + 4

So, the value of the integral of F(x) over the interval [0, π] is π3 + 4.

Example 2: – Integration of a Rational Function

Find the integral of the function f(x) = (x2 + 3x + 2) / (x + 2) dx.

Solution:

To integrate the rational function f(x), we can apply the technique of partial fractions decomposition.

First, we divide the numerator by the denominator:

X2 + 3x + 2 = (x + 1)(x + 2)

Now, we express the rational function as a sum of simpler fractions:

f(x) = (x2 + 3x + 2) / (x + 2) = (x + 1) + (1 / (x + 2))

Integrating each term separately, we have:

∫ f(x) dx = ∫ (x + 1) dx + ∫ (1 / (x + 2)) dx

Using the power rule of integration, the first term becomes:

= (x2 / 2 + x) + C1

The second term involves the natural logarithm function:

= ln(|x + 2|) + C2

So, the integral of the rational function f(x) is:

∫ f(x) dx = (x2 / 2 + x) + ln(|x + 2|) + C

 

Where C1 and C2 are constants of integration.

Key Points

  • Linearity: The integral of a sum or difference of functions is the sum or difference of their integrals.
  • Constant Multiple Rule: A constant factor can be pulled out of the integral.
  • Bounds: Integrals have upper and lower bounds that define the interval of integration.
  • Reversing Bounds: Reversing the bounds of integration changes the sign of the result.
  • Symmetric Functions: Integrating a symmetric function over a symmetric interval yields zero.
  • Odd Functions: Integrating an odd function over a symmetric interval also results in zero.
  • Periodic Functions: Integrating a periodic function over one period gives the same result for any period.
  • Absolute Values: Integrals involving absolute values can be split at points where the function changes sign.
  • Power Rule: The integral of x^n is (1/(n+1)) * x^(n+1) + C, where C is the constant of integration.
  • Substitution: Substituting a new variable can simplify the integral.
  • Partial Fractions: Decomposing a rational function into simpler fractions allows for easier integration.
  • Improper Integrals: Integrals with infinite limits or discontinuities are evaluated using limits.

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