# Untitled Page 21

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## 4.3 Properties of the integral

Let *f* and *g* be two functions of *x*, and let *c* be a constant. We already know that for derivatives,

But since the indefinite integral is just the operation of undoing a derivative, the same kind of rules must hold true for indefinite integrals as well:

And since a definite integral can be found by plugging in the upper and lower limits of integration into the indefinite integral, the same properties must be true of definite integrals as well.

### Example 5

◊ Evaluate the indefinite integral

◊ Using the additive property, the integral becomes

Then the property of scaling by a constant lets us change this to

We need a function whose derivative is *x*, which would be *x*^{2}/2, and
one whose derivative is sin *x*, which must be -cos *x*, so the result is