Some Properties of the Indefinite Integral

These are immediate consequences of the corresponding properties of derivatives. In each equation there is really an arbitrary constant of integration hanging around.

Let f (x) and g(x) be functions and $\lambda$ a constant. Then

$$\int \lambda \lambda \int$$

$$\int \int \int$$

provided that the integrals exist; we have not yet shown that there necessarily exists a function F(x) that differentiates to give a specified function f (x).

These rules may allow us to reduce an integral to the point where we can spot the answer.

$$\int$$2x + e^x dx = 2$$\int$$x dx + $$\int$$e^x dx = x² + e^x

$$\int$$sin x + 2 cos x dx = $$\int$$sin x dx + 2$$\int$$cos x dx = - cos x + 2 sin x