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An arbitrary constant should be added to each indefinate intergral. Indefinate intergrals can differ by constant. It is assumed that arguments of the logarithms are positive. |
(34) \( \int \sqrt{a^2 - v^2} dv = \) | ||
| (1) \( \int v^n dv = \frac{v^{n+1}}{{n+1}}\) where \( (n\neq-1) \) | (2) \( \int \frac{1}{{v}} dv = \int v^{-1} dv = ln(v) \) | (35) \( \int \sqrt{v^2 - a^2} dv = \) | |
| (3) \( \int \frac{1}{{av + b}} dv = \frac{1}{a} ln(av+b) \) where \( (a\neq 0) \) | (4) \( \int e^v dv = e^v \) , and \( \int e^{av+b} dv = \frac{1}{a} e^{av + b} \) | (36) \( \int \sqrt{a^2 + v^2} dv = \) | |
| (5)\( \int sin(av + b)^n dv = \frac{(av + b)^{n+1}}{a(n+1)} \) | (6)\( \int sin v dv = -cos v \) | (37) \( \int e^{f(v)f'(v) dv = e^{f(v)} } \) | (38) \( \int [f(v)]^n f'(v) dv = \frac{[f(v)]^{n+1}}{n+1} \) |
| (7)\( \int sin(av + b) dv = -\frac{1}{a} cos(av + b) \) where \( (a\neq 0) \) | (8)\( \int cos v dv = sin v \) | (39) \( \int \frac{f'(v)}{f(v)} dv = ln[f(v)] \) | (40) \( \int ln(v) dv = v[ln(v)-1] \) |
| (9)\( \int cos(av + b) dv = \frac{1}{a} sin(av + b) \) where \( (a\neq 0) \) | (10)\(\int tan v dv = -ln(cosv) = ln(secv)\) | (41) \( \int u \frac{dv}{dx} dx = uv - \int v\frac{du}{dx}dx \) | (42) \( \int e^{av}sin(bv) dv = \frac{e^{av}}{a^2 + b^2}[asin(bv) - bcos(bv)] \) |
| (11)\(\int cot v dv = ln(sinv) \) | (12)\(\int \) | or \( \int udv = uv - \int vdu \) | (43) \( \int e^{av}cos(bv) dv = \frac{e^{av}}{a^2 + b^2}[acos(bv) - bsin(bv)] \) |
| (13)\(\int sec v dv = ln(sec v + tan v) \) | (15)\(\int \) | ||
| (14)\(\int cosec v dv = - ln \) | (17)\(\int \) | (44) \( \int f(x) dx = \int f(g(u))\frac{dx}{du} du \) where | (45) \( \int_0^\infty e^{v^2} dv = \frac{\sqrt{\pi}}{2} \) |
| (16)\(\int sin^2v = \frac{1}{2}(v - sin v cos v) \) | (46) \( \int_{0}^{\frac{\pi}{2}} sin^nv dv = \frac{n-1}{n} \int_{0}^{\frac{\pi}{2}} sin^{n-2}vdv \) for any n | ||
| (18)\(\int cos^2v = \frac{1}{2}(v + sin v cos v) \) | (48) \(\int_{0}^{\frac{\pi}{2}} sin^mvcos^nv dv = I_{m,n} = \frac{m-1}{m+n} I_{m-2,n} = \frac{n-1}{m+n}I_{m,n-2} \) | ||
| (19)\(\int cosecv cotv dv = -cosecv \) | (20)\(\int sinhv dv = coshv \) | ||
| (21)\(\int coshv dv = sinh v\) | (22)\(\int sech^2 v dv = tanhv\) | ||
| (23)\(\int cosech^2v dv = ln(sinhv) \) | (24)\(\int tanh v = ln(coshv)\) | Numerical Intergration (a) Trapezium rule. One strip \( \int_{x_0}^{x_1} y dx = \frac{h}{2}[y_0 + y_1] + \epsilon \) |
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| (25)\(\int cothv dv = ln(shinhv) \) | (26)\(\int sechv dv = 2tan^{-1}(e^v)\) | ||
| (27)\(\int cosechv dv = ln (\frac{e^v-1}{e^v+1}) \) | (28)\(\int sechv tanhv dv = -sechv\) | (b) Simpson's rule. Two strip \( \int_x_0 \) | |
| (29)\(\int \frac{1}{a^2 + v^2} dv = \frac{1}{a}tan^{-1}(\frac{v}{a}) \) | (30)\(\int \frac{1}{\sqrt({a^2 - v^2})} dv = sin^{-1}\frac{v}{a} \) | ||
| (31)\(\int \frac{1}{a^2 - v^2} dv = \frac{1}{a}tanh^{-1} (\frac {v}{a}) \) | (c) Milne 4 strip \( \int_x_0 ydx = \frac{4h}{3} [2y_1 - y_2 - 2y_3] + \epsilon \) where | ||
| (32)\(\int \frac{1}{\sqrt{v^2 + a^2}} dv = sinh^{-1} (\frac{v}{a}) \) or \( ln[v + \sqrt{v^2 + a^2}] \) | |||
| (33)\(\int \frac{1}{\sqrt{a^2 - v^2}} dv = cosh^{-1} (\frac{v}{a}) \) or \( ln[v + \sqrt{v^2 - a^2}] \) |