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Property 1: Linearity of Definite Integration
- If f(x) and g(x) are integrable functions and c is a constant, then
- ∫abcf(x)+g(x):dx=c∫abf(x):dx+∫abg(x):dx
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Property 2: Change of Limits
- If f(x) is integrable on [a,b] and c and d are constants, then
- ∫aaf(x):dx=0
- ∫abf(x):dx=−∫baf(x):dx
- ∫abf(x):dx=∫acf(x):dx+∫cbf(x):dx
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Property 3: Definite Integral of a Constant
- If f(x)=c, a constant, then
- ∫abc:dx=c(b−a)
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Property 4: Additivity of Definite Integration
- If f(x) is integrable on [a,b] and c is a constant, then
- ∫abf(x):dx=∫acf(x):dx+∫cbf(x):dx
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Property 5: Conservation of Definite Integrals under Limits Transformation
- If f(x) is integrable on [a,b] and c and d are such that f(c+d−x) is integrable on [a,b], then
- ∫abf(x):dx=∫abf(c+d−x):dx