MA005: Calculus I

The following properties are true of all antiderivative

General Properties of Antiderivatives

If f and g are integrable functions, then

1. Constant Multiple $\int \mathrm{k} \cdot \mathrm{f}(x) \mathrm{dx}=\mathrm{k} \cdot \int \mathrm{f}(x) \mathrm{dx}$

2. Sum $\int \mathrm{f}(x)+\mathrm{g}(x) \mathrm{dx}=\int \mathrm{f}(x) \mathrm{dx}+\int \mathrm{g}(x) \mathrm{dx}$

3. Difference $\int \mathrm{f}(x)-\mathrm{g}(x) \mathrm{dx}=\int \mathrm{f}(x) \mathrm{dx}-\int \mathrm{g}(x) \mathrm{dx}$

There are general rules for derivatives of products and quotients. Unfortunately, there are no easy general patterns for antiderivatives of products and quotients, and only one more general property will be added to the list (in Chapter 6).

We have already found antiderivatives for a number of important functions

Particular Antiderivative

1. Constant Function: $\int \mathrm{k} \mathrm{d} \mathrm{x}=\mathrm{k} x+\mathrm{C}$

2. Powers of x: $\begin{aligned} &\int x^{\mathrm{n}} \mathrm{dx}=\frac{x^{\mathrm{n}+1}}{\mathrm{n}+1}+\mathrm{C} \quad \text { if } \mathrm{n} \neq-1 \\ &\int x^{-1} \mathrm{dx}=\int \frac{1}{x} \mathrm{~d} \mathrm{x}=\ln |x|+\mathrm{C} \end{aligned}$

Common special cases: $\begin{aligned} &\int \sqrt{x} \mathrm{dx}=\frac{2}{3} x^{3 / 2}+\mathrm{C} \\ &\int \frac{1}{\sqrt{x}} \mathrm{dx}=2 x^{1 / 2}+\mathrm{C}=2 \sqrt{x}+\mathrm{C} \end{aligned}$

3. Trigonometric Functions: $\begin{aligned} &\int \cos (x) \mathrm{dx}=\sin (x)+\mathrm{C} \quad \int \sin (x) \mathrm{d} \mathrm{x}=-\cos (x)+\mathrm{C} \\ &\int \sec ^{2}(x) \mathrm{d} \mathrm{x}=\tan (x)+\mathrm{C} \quad \int \csc ^{2}(x) \mathrm{d} x=-\cot (x)+\mathrm{C} \\ &\int \sec (x) \cdot \tan (x) \mathrm{dx}=\sec (x)+\mathrm{C} \quad \int \csc (x) \cdot \cot (x) \mathrm{dx}=-\csc (x)+\mathrm{C} \end{aligned}$

4. Exponential Functions: $\int \mathrm{e}^{x} \mathrm{dx}=\mathrm{e}^{x}+\mathrm{C}$

All of these antiderivatives can be verified by differentiating. The list of antiderivatives of particular functions will continue to grow in the coming chapters and will eventually include antiderivatives of additional trigonometric functions, the inverse trigonometric functions, logarithms, and others.