Properties of Indefinite Integrals

We will now look at some properties of indefinite integrals, all of which are analogous to properties of definite integrals.

Theorem 1: If $\int f(x) \: dx$ and $\int g(x) \: dx$ exist, and $k \in \mathbb{R}$, then: a) $\int f(x) \: dx + \int g(x) \: dx = \int f(x) + g(x) \: dx$ (Addition Property). b) $\int f(x) \: dx - \int g(x) \: dx = \int f(x) - g(x) \: dx$ (Subtraction Property). c) $k \cdot \int f(x) \: dx = \int kf(x) \: dx$. (Multiple Property). d) If $F(x) = \int f(x) \: dx$, then $\frac{d}{dx} F(x) = f(x)$.

We will now look at some examples of applying indefinite integral properties.

Example 1

Evaluate $\int x^2 + e^x \: dx$.

We note that $\frac{d}{dx} \frac{x^3}{3} = x^2$ and $\frac{d}{dx} e^x = e^x$, and so $\int x^2 + e^x \: dx = \frac{x^3}{3} + e^x + C$.

Example 2

Simplify the following expression: $b \int f(x) \: dx - c \int g(x) \: dx + \frac{d}{dx} (c^2 \int m(x) \: dx)$.

By applying all three properties above we obtain:

(1)

\begin{align} \: b \int f(x) \: dx - c \int g(x) \: dx + \frac{d}{dx} ( c^2 \int m(x) \: dx )\\ = \int bf(x) \: dx - \int cg(x) \: dx + c^2 \frac{d}{dx} \int m(x) \: dx \\ = \int bf(x) - cg(x) \: dx + c^2m(x) \end{align}