Integrals

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Integration is the inverse process of differentiation. While differentiation finds the rate of change of a function, integration finds the original function when its derivative is given. If f'(x) = g(

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An anti-derivative (or primitive) of a function f(x) is a function F(x) such that F'(x) = f(x). Example: Since d/dx(sin x) = cos x, therefore sin x is an anti-derivative of cos x. Generally, sin x + C

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The symbol is ∫f(x)dx. Here: ∫ is the integral sign, f(x) is the integrand (function to be integrated), x is the variable of integration, and dx indicates integration with respect to x. The result is

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The constant C appears because differentiation of any constant is zero. If F(x) is an anti-derivative of f(x), then F(x) + C is also an anti-derivative for any constant C, since d/dx[F(x) + C] = F'(x)

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∫x^n dx = x^(n+1)/(n+1) + C, where n ≠ -1. This is valid for all real numbers n except -1. For n = -1, we have ∫x^(-1) dx = ∫(1/x) dx = log|x| + C.

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∫cos x dx = sin x + C ∫sin x dx = -cos x + C ∫sec²x dx = tan x + C ∫cosec²x dx = -cot x + C ∫sec x tan x dx = sec x + C ∫cosec x cot x dx = -cosec x + C

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∫1/√(1-x²) dx = sin⁻¹x + C = -cos⁻¹x + C ∫1/(1+x²) dx = tan⁻¹x + C ∫1/(x√(x²-1)) dx = sec⁻¹x + C (for |x| > 1)

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∫e^x dx = e^x + C ∫a^x dx = a^x/log a + C (where a > 0, a ≠ 1) ∫1/x dx = log|x| + C Note: The absolute value in log|x| is important for x < 0.