Integrating Basic Algebraic and Trigonometric Functions

\(\displaystyle \int \bigl(k f\bigr) = k \int f \text{ for } k \in \mathbf{R}\)
\(\displaystyle \int \bigl(f \pm g\bigr) = \int f \pm \int g \)
\(\displaystyle \int \Bigl((f' \circ g)(g') \Bigr) = (f \circ g) + C \)
\(\displaystyle \int f g \mathbf{\neq} \int f + \int g \)
\(\displaystyle \int \frac{f}{g} \mathbf{\neq} \frac{\int f}{\int g} \)
\(\displaystyle \int x^n \,\mathrm{d}x = \frac{1}{n+1}x^{n+1} + C \text{ for } n\neq -1\)
\(\displaystyle \int \cos(x) \,\mathrm{d}x = \sin(x) + C \)
\(\displaystyle \int \sec(x)\tan(x) \,\mathrm{d}x = \sec(x) + C \)
\(\displaystyle \int \sec^2(x) \,\mathrm{d}x = \tan(x) + C\)
\(\displaystyle \int \sin(x) \,\mathrm{d}x = -\cos(x) + C \)
\(\displaystyle \int \csc(x)\cot(x) \,\mathrm{d}x = -\csc(x) + C \)
\(\displaystyle \int \csc^2(x) \,\mathrm{d}x = -\cot(x) + C\)

For each of these indefinite integrals, write down a formula for the family of all antiderivatives of its integrand. Use software capable of symbolic integration to check your formulas.

\(\displaystyle \int 17x^2-2 \,\mathrm{d}x \)
\(\displaystyle \int x^{17} \,\mathrm{d}x \)
\(\displaystyle \int \frac{1}{x^{17}} \,\mathrm{d}x \)
\(\displaystyle \int \sqrt[17]{x} \,\mathrm{d}x \)
\(\displaystyle \int \frac{1}{17} \,\mathrm{d}x \)
\(\displaystyle \int 17x^{\pi}\,\mathrm{d}x \)
\(\displaystyle \int 1.7x^{\pi}\,\mathrm{d}x \)
\(\displaystyle \int \frac{1}{(x\pi)^2}\,\mathrm{d}x \)
\(\displaystyle \int 1.7^{-1}x^{7.1}\,\mathrm{d}x \)
\(\displaystyle \int 1.7^{2}x^{-7.1}\,\mathrm{d}x \)
\(\displaystyle \int x^{2}x^{-7.1}\,\mathrm{d}x \)
\(\displaystyle \int x^{2}17^{-7.1}\,\mathrm{d}x \)
\(\displaystyle \int x^3+x^2+x+1+\frac{1}{x^2}+\frac{1}{x^3}\,\mathrm{d}x \)
\(\displaystyle \int \bigl(x-17\bigr)^2 \,\mathrm{d}x \)
\(\displaystyle \int \bigl(x-17^2\bigr) \,\mathrm{d}x \)
\(\displaystyle \int \bigl(x-2\bigr)^{17} \,\mathrm{d}x \)
\(\displaystyle \int \bigl(17x-2\bigr)^{17} \,\mathrm{d}x \)
\(\displaystyle \int x\bigl(17x^2-2\bigr)^{17} \,\mathrm{d}x \)
\(\displaystyle \int \bigl(17x^{16}-17\bigr)\bigl(x^{17}-17x+2\bigr) \,\mathrm{d}x \)
\(\displaystyle \int \frac{x^{16}-1}{(x^{17}-17x+2)^2} \,\mathrm{d}x \)
\(\displaystyle \int \frac{1}{\sqrt[7]{x}} \,\mathrm{d}x \)
\(\displaystyle \int x\sqrt{x^2-1} \,\mathrm{d}x \)
\(\displaystyle \int x^3\sqrt{x^2-1} \,\mathrm{d}x \)
\(\displaystyle \int x^7\sqrt{x^2-1} \,\mathrm{d}x \)
\(\displaystyle \int (x-1)\sqrt{x^2-2x-1} \,\mathrm{d}x \)
\(\displaystyle \int \frac{(x-1)}{\sqrt{x^2-2x-1}} \,\mathrm{d}x \)
\(\displaystyle \int 71\sin(x) + 17\cos(x) \,\mathrm{d}x \)
\(\displaystyle \int \sec(x)\tan(x) \,\mathrm{d}x \)
\(\displaystyle \int 17\sin(x)\cos(x) \,\mathrm{d}x \)
\(\displaystyle \int 17(2x-1)\sec(x^2-x-1)\tan(x^2-x-1) \,\mathrm{d}x \)
\(\displaystyle \int \sin(x)\sin\bigl(\cos(x)\bigr) \,\mathrm{d}x \)
\(\displaystyle \int \cos(x)\sin^{3}(x) \,\mathrm{d}x \)
\(\displaystyle \int 5x^6\sec^2(x^7) \,\mathrm{d}x \)
\(\displaystyle \int \tan(x)\tan\bigl(\sec(x)\bigr)\sec(x) \,\mathrm{d}x \)