Integrating Basic Algebraic and Trigonometric Functions

\(\displaystyle \int \bigl(k f\bigr) = k \int f \text{ for } k \in \mathbf{R}\)

\(\displaystyle \int \bigl(f \pm g\bigr) = \int f \pm \int g \)

\(\displaystyle \int \Bigl((f' \circ g)(g') \Bigr) = (f \circ g) + C \)

\(\displaystyle \int f g \mathbf{\neq} \int f + \int g \)

\(\displaystyle \int \frac{f}{g} \mathbf{\neq} \frac{\int f}{\int g} \)

\(\displaystyle \int x^n \,\mathrm{d}x = \frac{1}{n+1}x^{n+1} + C \text{ for } n\neq -1\)

\(\displaystyle \int \cos(x) \,\mathrm{d}x = \sin(x) + C \)

\(\displaystyle \int \sec(x)\tan(x) \,\mathrm{d}x = \sec(x) + C \)

\(\displaystyle \int \sec^2(x) \,\mathrm{d}x = \tan(x) + C\)

\(\displaystyle \int \sin(x) \,\mathrm{d}x = -\cos(x) + C \)

\(\displaystyle \int \csc(x)\cot(x) \,\mathrm{d}x = -\csc(x) + C \)

\(\displaystyle \int \csc^2(x) \,\mathrm{d}x = -\cot(x) + C\)

Each of the following expressions of a variable \(x\) can be thought of as the formula for a real-valued function that’s continuous on some open subset of \(\mathbf{R}.\) As some healthy exercise, for each of these functions, write down a general formula for the family of all of its antiderivatives — i.e. its indefinite integral.

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