Page 33 - Calc for the Clueless

The fundamental theorem of integral calculus. If f(x) is continuous on [a,b] and F(x) is any antiderivative of
f(x), then

Example 26—

Note

dx, the definite integral, is a number if a and b are numbers. dx the antiderivative or indefinite
integral, is a family of functions, each of which differs from the others by a constant.

We rewrite previous rules using the indefinite integral.

Example 27—

Example 28—

This is a change of variables. We let u equal whatever is in parentheses, under a radical sign,
2
3
2
2
etc. Let u = x + 1. du/dx = 3x . Solve for dx. du = 3x dx. So dx = du/3x .

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The limits must change. u = x + 1. x = 0. u = 3 + 1 = 1.
3
x = 3. u = 3 + 1 = 28.
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