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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,

\[ \frac{d}{dx}(f+g) = \frac{df}{dx} + \frac{dg}{dx} \]

and

\[ \frac{d}{dx}(cf) = c\frac{df}{dx} \]

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:

\[ \int(f+g)dx = \int f dx + \int g dx \]

and

\[ \int(cf)dx = c \int f dx \]

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 54

◊ Evaluate the indefinite integral

\[ \int (x+2\sin x) dx . \]

◊ Using the additive property, the integral becomes

\[ \int x dx + \int 2\sin x dx . \]

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

\[ \int x dx + 2 \int \sin x dx . \]

We need a function whose derivative is x, which would be x2/2, and one whose derivative is sin x, which must be -cos x, so the result is

\[ \frac{1}{2}x^2 - 2 \cos x + c . \]