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Definition

If f and F are functions of x such that F′(x) = f(x) then the function F is called a primitive or an indefinite integral of f with respect to x and is expressed as

where c is a constant of integration.

f(x) then possesses infinitely many antiderivatives, all of them being contained in the expression F(x) + c where c is a constant.

From the geometric point of view, an indefinite integral is a collection (family) of curves, each of which is obtained by translating one of the curves parallel to itself upwards or downwards.

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Expression Substitution

x = a sin θ or a cos θ

x = a tan θ or a cot θ

x = a sec θ or a cosec θ

x = a cos θ or a cos 2 θ

x = a (1 - cos θ)

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Critical Points

The points on the curve y = f(x) at which dy/dx = 0 or dy/dx does not exist are called critical points.