Definite Integrals, Formula, Properties, Rules, Integration over Discontinuities - Calculus

Definite integral properties in calculus simplify evaluating integrals and calculating the net signed area under a curve. Key properties include: the integral is zero when limits are the same (\int_a^a f(x)dx = 0), reversing limits changes the sign (\int_a^b f(x)dx = -\int_b^a f(x)dx), an integral of a sum is the sum of integrals (\int_a^b [f(x) \pm g(x)]dx = \int_a^b f(x)dx \pm \int_a^b g(x)dx), a constant factor can be pulled out (\int_a^b kf(x)dx = k\int_a^b f(x)dx), and an integral can be split into two intervals (\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx).

💡Basic Properties
• Zero Integral: If the upper and lower limits of integration are the same, the definite integral is zero.
◦ Formula: \int_a^a f(x)dx = 0
◦ Explanation: The interval has no width, and thus no area is enclosed.
• Reverse Limits: If you swap the upper and lower limits of integration, the sign of the definite integral reverses.
◦ Formula: \int_a^b f(x)dx = -\int_b^a f(x)dx
◦ Explanation: This signifies a change in the accumulated area's direction.

💡Properties for Calculation
• Sum/Difference of Functions: The integral of a sum or difference of functions is the sum or difference of their individual integrals.
◦ Formula: \int_a^b [f(x) \pm g(x)]dx = \int_a^b f(x)dx \pm \int_a^b g(x)dx
◦ Explanation: This property allows you to break down complex integrals into simpler ones.
• Constant Multiple: A constant factor multiplied by a function within an integral can be factored out.
◦ Formula: \int_a^b kf(x)dx = k\int_a^b f(x)dx (where 'k' is a constant)
◦ Explanation: Constants can be pulled out of the integral to simplify the integration process.
• Additive Interval Property: An integral over an interval can be split into the sum of two integrals over sub-intervals.
◦ Formula: \int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx
◦ Explanation: This property is valid for any value of 'c', not just when it lies between 'a' and 'b'.

💡Other Useful Properties
• Variable Substitution: A definite integral's value is independent of the variable of integration.
◦ Formula: \int_a^b f(x)dx = \int_a^b f(t)dt
◦ Explanation: You can replace the variable of integration with another variable, like 't', without changing the outcome.
• Integration of Even and Odd Functions: Properties exist for integrating even or odd functions over symmetric intervals around the origin, which can simplify calculations.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link
• Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link

💡Chapters:
00:00 Definite integrals, properties
01:35 Integrals over discontinuities

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