16.9: Derivatives and Primitives (Indefinite Integrals)
- Page ID
- 107068
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)| \(f(x)\) | \(f'(x)\) | \(\int f(x)dx\)(\(\pm c\)) |
|---|---|---|
| \(k\) | 0 | \(kx\) |
| \(x^n\) | \(n x^{n-1}\), \(n \neq 0\) | \(\frac{x^{n+1}}{n+1}\), \(n \neq -1\) |
| \(\frac{1}{x}\) | \(-\frac{1}{x^2}\) | \(ln|x|\) |
| \(a^x\) | \(a^x \ln{a}\) | \(\frac{a^x}{\ln{a}}\) |
| \(e^x\) | \(e^x\) | \(e^x\) |
| \(\log_a{x}\) | \(\frac{1}{x \ln{a}}\) | \(\frac{x \ln{x}-x}{\ln{a}}\) |
| \(\ln{x}\) | \(\frac{1}{x}\) | \(x \ln{x}-x\) |
| \(\sin{x}\) | \(\cos{x}\) | \(-\cos{x}\) |
| \(\cos{x}\) | \(-\sin{x}\) | \(\sin{x}\) |
| \(\tan{x}\) | \(\frac{1}{\cos^2{x}}\) | \(-\ln{(\cos{x}})\) |
| \(\arcsin{x}\) | \(\frac{1}{\sqrt{1-x^2}}\) | \(x \arcsin{x}+\sqrt{1-x^2}\) |
| \(\arccos{x}\) | \(-\frac{1}{\sqrt{1-x^2}}\) | \(x \arccos{x}-\sqrt{1-x^2}\) |
| \(\arctan{x}\) | \(\frac{1}{1+x^2}\) | \(x \arctan{x}-\frac{1}{2}\ln{(1+x^2)}\) |
|
\(\frac{1}{a^2+x^2}\) |
\(\frac{-2x}{(a^2+x^2)^2}\) | \(\frac{1}{a}\arctan{\left( \frac{x}{a}\right)}\) |
|
\(\frac{1}{\sqrt{a^2-x^2}}\) |
\(\frac{x}{(a^2-x^2)^{\frac{3}{2}}}\) | \(\arcsin{\left( \frac{x}{a}\right)}\) |
- \(\int \sin^2{(ax)}dx= \frac{x}{2}-\frac{\sin{(2ax)}}{4a}+c\)
- \(\int \cos^2{(ax)}dx= \frac{x}{2}+\frac{\sin{(2ax)}}{4a}+c\)
- \(\int \sin^3{(ax)}dx= \frac{1}{12a}\cos{(3ax)}-\frac{3}{4a}\cos{(ax)}+c\)
- \(\int \cos^3{(ax)}dx= \frac{1}{12a}\sin{(3ax)}+\frac{3}{4a}\sin{(ax)}+c\)
- \(\int x \cos{(ax)}dx=\frac{\cos{(ax)}}{a^2}+\frac{\sin{(ax)}}{a}x+c\)
- \(\int x \sin{(ax)}dx=\frac{\sin{(ax)}}{a^2}-\frac{\cos{(ax)}}{a}x+c\)
- \(\int x \sin^2{(ax)}dx=\frac{x^2}{4}-\frac{x\sin{(2ax)}}{4a}-\frac{\cos{(2ax)}}{8a^2}+c\)
- \(\int xe^{x^2}dx=e^{x^{2}}/2+c\)
- \(\int x e^{ax}=\frac{e^{ax}(ax-1)}{a^2}+c\)
- \(\int \frac{x}{x^2+1}dx=\frac{1}{2}\ln{(1+x^2)} +c\)