16.9: Derivatives and Primitives (Indefinite Integrals)
- Page ID
- 107068
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| \(f(x)\) | \(f'(x)\) | \(\int f(x)dx\)(\(\pm c\)) |
|---|---|---|
| \(k\) | 0 | \(kx\) |
| \(x^n\) | \(n x^{n-1}\), \(n \neq 0\) | \(\frac{x^{n+1}}{n+1}\), \(n \neq -1\) |
| \(\frac{1}{x}\) | \(-\frac{1}{x^2}\) | \(ln|x|\) |
| \(a^x\) | \(a^x \ln{a}\) | \(\frac{a^x}{\ln{a}}\) |
| \(e^x\) | \(e^x\) | \(e^x\) |
| \(\log_a{x}\) | \(\frac{1}{x \ln{a}}\) | \(\frac{x \ln{x}-x}{\ln{a}}\) |
| \(\ln{x}\) | \(\frac{1}{x}\) | \(x \ln{x}-x\) |
| \(\sin{x}\) | \(\cos{x}\) | \(-\cos{x}\) |
| \(\cos{x}\) | \(-\sin{x}\) | \(\sin{x}\) |
| \(\tan{x}\) | \(\frac{1}{\cos^2{x}}\) | \(-\ln{(\cos{x}})\) |
| \(\arcsin{x}\) | \(\frac{1}{\sqrt{1-x^2}}\) | \(x \arcsin{x}+\sqrt{1-x^2}\) |
| \(\arccos{x}\) | \(-\frac{1}{\sqrt{1-x^2}}\) | \(x \arccos{x}-\sqrt{1-x^2}\) |
| \(\arctan{x}\) | \(\frac{1}{1+x^2}\) | \(x \arctan{x}-\frac{1}{2}\ln{(1+x^2)}\) |
|
\(\frac{1}{a^2+x^2}\) |
\(\frac{-2x}{(a^2+x^2)^2}\) | \(\frac{1}{a}\arctan{\left( \frac{x}{a}\right)}\) |
|
\(\frac{1}{\sqrt{a^2-x^2}}\) |
\(\frac{x}{(a^2-x^2)^{\frac{3}{2}}}\) | \(\arcsin{\left( \frac{x}{a}\right)}\) |
- \(\int \sin^2{(ax)}dx= \frac{x}{2}-\frac{\sin{(2ax)}}{4a}+c\)
- \(\int \cos^2{(ax)}dx= \frac{x}{2}+\frac{\sin{(2ax)}}{4a}+c\)
- \(\int \sin^3{(ax)}dx= \frac{1}{12a}\cos{(3ax)}-\frac{3}{4a}\cos{(ax)}+c\)
- \(\int \cos^3{(ax)}dx= \frac{1}{12a}\sin{(3ax)}+\frac{3}{4a}\sin{(ax)}+c\)
- \(\int x \cos{(ax)}dx=\frac{\cos{(ax)}}{a^2}+\frac{\sin{(ax)}}{a}x+c\)
- \(\int x \sin{(ax)}dx=\frac{\sin{(ax)}}{a^2}-\frac{\cos{(ax)}}{a}x+c\)
- \(\int x \sin^2{(ax)}dx=\frac{x^2}{4}-\frac{x\sin{(2ax)}}{4a}-\frac{\cos{(2ax)}}{8a^2}+c\)
- \(\int xe^{x^2}dx=e^{x^{2}}/2+c\)
- \(\int x e^{ax}=\frac{e^{ax}(ax-1)}{a^2}+c\)
- \(\int \frac{x}{x^2+1}dx=\frac{1}{2}\ln{(1+x^2)} +c\)