### 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

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