Properties Of Integrals & Technique

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.

Most Read

 Unit: Infinite Sequence & Series Chapter: Taylor Series, Maclaurin Series Reference: – Divergence test, Geometric series, Integral test, Comparison test, Limit comparison test, Alternating series test, Absolute convergence, Conditional Convergence, Ratio test, Root test, Taylor series, Radius of convergence, Interval of convergence.   After studying this chapter, you should be able to: Introduction to Taylor […]

Unit: Infinite Sequence & Series Chapter: Error Bound & Interval of Series Reference: – Taylor Series, Remainder term, Lagrange error Bound, Alternating series error bound, Integral test, Comparison test, Ratio test, Root test, Geometric series, Power series, Maclaurin series, Binomial series, Estimating series sums, Interval estimation techniques.   After studying this chapter, you should be […]

 Unit: Infinite Sequence & Series Chapter: Divergence & Convergence Tests Reference: – Divergence test, Geometric series, Integral test, Comparison test, Limit comparison test, Alternating series test, Absolute convergence, Conditional Convergence, Ratio test, Root test, Taylor series, Radius of convergence, Interval of convergence. After studying this chapter, you should be able to: Introduction to Convergence & […]