{"id":10138,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=10138"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"properties-of-integrals-technique","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/properties-of-integrals-technique\/","title":{"rendered":"Properties Of Integrals & Technique"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit: <\/strong>Integration & Accumulation of Change<\/strong><\/h2>\n

Chapter: <\/strong>Properties of Integrals & Techniques<\/strong><\/h3>\n

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.<\/em><\/p>\n

After studying this chapter, you should be able to:<\/strong><\/p>\n

    \n
  • Linearity of Integrals & Constant multiple rules.<\/li>\n
  • Bounds, Limits & Reversing the Bounds.<\/li>\n
  • Integrals of Symmetric & Odd Functions.<\/li>\n
  • Integrals of Periodic Functions & Absolute Values.<\/li>\n<\/ul>\n

     <\/p>\n

    Linearity of Integration & Constant Multiple Rules<\/u><\/strong><\/p>\n

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

          Bounds, Limits & Reversing the Bounds:<\/u><\/strong><\/p>\n

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

                   <\/p>\n

                  Integrals of Symmetric & Odd functions<\/u><\/strong>: –<\/strong><\/p>\n

                   <\/p>\n

                    \n
                  1. Integrals of Symmetric Functions<\/u>:\n
                      \n
                    • 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.<\/li>\n
                    • When integrating a symmetric function over an interval symmetric about the y-axis, the result is zero.<\/li>\n
                    • Mathematically, if f(x) is a symmetric function and the interval of integration is [-a, a], then: ∫[-a, a] f(x) dx = 0.<\/li>\n
                    • This property arises because the positive and negative areas on either side of the y-axis cancel each other out.<\/li>\n<\/ul>\n<\/li>\n
                    • Integrals of Odd Functions<\/u>:\n
                        \n
                      • An odd function is symmetric about the origin (0, 0), meaning that for every value of x in its domain, f(x) = -f(-x).<\/li>\n
                      • When integrating an odd function over an interval symmetric about the origin, the result is also zero.<\/li>\n
                      • Mathematically, if f(x) is an odd function and the interval of integration is [-a, a], then: ∫[-a, a] f(x) dx = 0.<\/li>\n
                      • This property arises because the positive and negative areas on either side of the origin cancel each other out.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

                         <\/p>\n

                        Integration of Rational Function & Improper Integrals<\/u><\/strong>:<\/p>\n

                         <\/p>\n

                          \n
                        1. Integration of Rational Functions<\/u>:\n
                            \n
                          • 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.<\/li>\n
                          • To integrate a rational function, various techniques are used, including polynomial long division, partial fractions decomposition, and substitution.<\/li>\n
                          • 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.<\/li>\n
                          • Partial fractions decomposition is a method used to decompose a rational function into simpler fractions, making it easier to integrate.<\/li>\n
                          • After decomposition, the resulting fractions can be integrated using basic integration rules.<\/li>\n
                          • Substitution is often employed to simplify the integral by replacing a variable with a new variable or expression.<\/li>\n<\/ul>\n<\/li>\n
                          • Improper Integrals<\/u>:\n
                              \n
                            • An improper integral is an integral with infinite limits or an integrand that has discontinuities within the interval of integration.<\/li>\n
                            • Improper integrals are evaluated by taking limits as one or both of the bounds approach infinity or a point of discontinuity.<\/li>\n
                            • 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.<\/li>\n
                            • To evaluate improper integrals, techniques such as limit evaluation, breaking the integral into several parts, or using comparison tests are applied.<\/li>\n
                            • The limits involved in improper integrals ensure that the integral converges to a finite value or diverges to infinity.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

                              Integration of Periodic Function & Absolute values<\/u><\/strong><\/p>\n

                                \n
                              1. Integration of Periodic Functions<\/u>:\n
                                  \n
                                • A periodic function is a function that repeats itself after a certain interval called a period.<\/li>\n
                                • When integrating a periodic function over one period, the result is equal to the integral over any other period.<\/li>\n
                                • 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.<\/li>\n
                                • This property arises from the fact that the positive and negative areas of the periodic function cancel each other out over one period.<\/li>\n
                                • It allows us to evaluate the integral of a periodic function by considering a single period, simplifying the calculation.<\/li>\n<\/ul>\n<\/li>\n
                                • Integration of Absolute Values<\/u>:\n
                                    \n
                                  • 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.<\/li>\n
                                  • 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.<\/li>\n
                                  • 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.<\/li>\n
                                  • 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.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

                                    Example<\/u><\/strong>: – Using the Linearity of Integrals and Constant Multiple Rule<\/p>\n

                                     <\/p>\n

                                    Find the integral of the function F(x) = 3x2<\/sup> + 2sin(x) – 4cos(x) over the interval [0, π].<\/p>\n

                                    Solution<\/u><\/strong>:<\/p>\n

                                    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.<\/p>\n

                                     <\/p>\n

                                    ∫[0, π] F(x) dx = ∫[0, π] (3x2<\/sup>) dx + ∫[0, π] (2sin(x)) dx – ∫[0, π] (4cos(x)) dx<\/p>\n

                                    Using the power rule of integration, the first term becomes:<\/p>\n

                                    = [x3<\/sup>] from 0 to π<\/p>\n

                                    = π3<\/sup> – 0<\/p>\n

                                    = π3<\/sup><\/p>\n

                                    The second term involves integrating sin(x), which is a known function:<\/p>\n

                                    = [-2cos(x)] from 0 to π<\/p>\n

                                    = -2cos(π) + 2cos(0)<\/p>\n

                                    = -2(-1) + 2(1)<\/p>\n

                                    = 4<\/p>\n

                                    Similarly, for the third term involving cos(x), we have:<\/p>\n

                                    = [-4sin(x)] from 0 to π<\/p>\n

                                    = -4sin(π) + 4sin(0)<\/p>\n

                                    = -4(0) + 4(0)<\/p>\n

                                    = 0<\/p>\n

                                    Summing up the individual integrals, we get:<\/p>\n

                                    ∫[0, π] F(x) dx = π3<\/sup> + 4 + 0<\/p>\n

                                    = π3<\/sup> + 4<\/p>\n

                                    So, the value of the integral of F(x) over the interval [0, π] is π3<\/sup> + 4.<\/p>\n

                                    Example 2<\/u><\/strong>: – Integration of a Rational Function<\/p>\n

                                    Find the integral of the function f(x) = (x2<\/sup> + 3x + 2) \/ (x + 2) dx.<\/p>\n

                                    Solution<\/u><\/strong>:<\/p>\n

                                    To integrate the rational function f(x), we can apply the technique of partial fractions decomposition.<\/p>\n

                                    First, we divide the numerator by the denominator:<\/p>\n

                                    X2<\/sup> + 3x + 2 = (x + 1)(x + 2)<\/p>\n

                                    Now, we express the rational function as a sum of simpler fractions:<\/p>\n

                                    f(x) = (x2<\/sup> + 3x + 2) \/ (x + 2) = (x + 1) + (1 \/ (x + 2))<\/p>\n

                                    Integrating each term separately, we have:<\/p>\n

                                    ∫ f(x) dx = ∫ (x + 1) dx + ∫ (1 \/ (x + 2)) dx<\/p>\n

                                    Using the power rule of integration, the first term becomes:<\/p>\n

                                    = (x2<\/sup> \/ 2 + x) + C1<\/p>\n

                                    The second term involves the natural logarithm function:<\/p>\n

                                    = ln(|x + 2|) + C2<\/p>\n

                                    So, the integral of the rational function f(x) is:<\/p>\n

                                    ∫ f(x) dx = (x2<\/sup> \/ 2 + x) + ln(|x + 2|) + C<\/p>\n

                                     <\/p>\n

                                    Where C1 and C2 are constants of integration.<\/p>\n

                                    Key Points<\/u><\/strong><\/p>\n

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

                                      <\/p>\n