INTEGRATION CONCEPT
Integral = Anti derivatives = Primitive
Integration –The process of finding the function f(x) whose differential coeffiicient w.r.t. ‘x’, denoted by F (x) is given,
written as:
∫ F(x) dx = f(x) + C
Thus, integration is an inverse process of differentiation
or
integration is anti of differentiation. .
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Note:
●.All functions are not integrable and
●.the integral of a function is not unique.
Ex: √sinx , √cosx.
●. If a polynomial function of a degree n isintegrated we get a polynomial of degree n + 1.
An important fact.
d/dx [ ∫ f(x) dx = f(x) + C
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1. Integration is an operation on function.
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2 .∫ [ f(x) +g(x).... ] dx = ∫ f(x) dx + ∫g(x)dx....
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3. ∫ k dx = k∫ 1 dx + c= kx + C (Constant)
∫ 0 dx = C (Constant)
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4. ∫ xn dx = (xn+1) / (n+1),(where n ≠ -1)
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5. ∫ ex dx = ex + C.
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6. ∫ 1/x dx = log|x| + c
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8. ∫ sin x dx = – cos x + c
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9. ∫ cos x dx = sin x + c
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10. ∫tan x dx = log |sec x| + c
= – log |cos x | + c
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11. ∫ cot x dx = log |sin x| + c
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11. ∫ sec x dx = log |sec x + tan x| + c
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12.∫cosec x dx =log|cosec x – cot x|+ c
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Only sec²x, cosec²x has integration other are done after making changes.
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13. ∫ sin²x dx..........
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14. ∫ cos²x dx........
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15. ∫ tan²x dx.....
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16. ∫ cot²x dx......
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17. ∫ sec²x dx = tan x + c
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18. ∫ cosec²x dx = – cot x + c
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19. ∫ sec x tan x dx = sec x + c
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20. ∫ cosec x cot x dx = – cosec x + c
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Worksheet:
find the integration of sin²x , sin³x , sin⁴x , cos²x , cos³x , cos⁴x,
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जब Numerator 1 हो तथा Denominator under root के अंदर न आया हो, without root हो,
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जब Numerator 1 हो तथा Denominator under root के अंदर आया हो,
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