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.
◊ 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 x2/2, and one whose derivative is sin x, which must be -cos x, so the result is