Integrals class 12 of Class 12


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 ∫ f(x) dx = F(x) + c 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.

Expression Substitution

√a2 – x2 x = a sinθ or a conθ

√a2 – x2 x = a tanθ or a cotθ

√a2 – x2 x = a tanθ or a cotθ

√a2 – x2 x = a secθ or a cosecθ

√a+x/a-x or √a-x/a+x x = a conθ or a cos2θ

√2ax – x2 x = a (1 – cosθ)

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.

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