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Indefinite Integral

Given a function f, if F is a function such that

F ‘ (x) = f (x).

then F is called an antiderivative of f. Thus an at derivative of f is simply a function whose derivative is f.

For example since d/dx (x3) = 3x2, x3 is an antiderivative of 3x2. However, it is not the only antiderivative of 3x2.

Since d/dx (x3 + 2) = 3x2 and d/dx (x3- 7) = 3x2,

both x3 + 2 and x3 – 7 are also antiderivatives of 3x2. It can be shown that any antiderivative of 3x2 must have the form x3 + c. In general if F (X) is an antiderivative of f (x), then for any constant, we have

d/dx [F (x) + c] = d/dx (F (x) + 0 = f (x).

Therefore, F(x) + c is also an antiderivative of f(x).

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