Calculus Made Easy · Silvanus P. Thompson
(d y)/(d x) longleftarrow y longrightarrow ∫ y d x
Algebraic.
1 x 1/2 x^2+C
0 a ax+C
1 x ± a 1/2 ± ax^2+C
a ax 1/2 ± ax^2+C
2x x^2 1/3 x^3+C
n x^(n-1) x^n 1/(n+1) x^(n+1)+C
-x^(-2) x^(-1) log _(ε) x+C
(d u)/(d x) ± (d v)/(d x) ± (d w)/(d x) u ± v ± w ∫ u d x ± ∫ v d x ± ∫ w d x
u (d v)/(d x)+v (d u)/(d x) u v No general form known
(v (d u)/(d x)-u (d v)/(d x))/(v^2) u/v No general form known
(d u)/(d x) u u x-∫ x du+C
Exponential and Logarithmic.
ε^x ε^x ε^x+C
x^(-1) log _(ε) x x(log _(ε) x-1)+C
0.4343 × x^(-1) log _(10) x 0.4343 x(log _(ε) x-1)+C
a^x log _(ε) a a^x (a^x)/(log _(ε) a)+C
Trigonometrical.
cos x sin x - cos x+C
- sin x cos x sin x+C
sec ^2 x tan x - log _(ε) cos x+C
Circular (Inverse).
1/(√((1 - x^2)) arcsin x x · arcsin x+√(1-x^2)+C
-1/(√((1 - x^2)) arccos x x · arcsin x-√(1-x^2)+C
1/(1+x^2) arctan x x · arctan x-1/2 log _(ε)(1+x^2)+C
(d y)/(d x) longleftarrow y longrightarrow ∫ y d x
Hyperbolic.
cosh x sinh x cosh x + C
sinh x cosh x sinh x + C
sech ^2 x tanh x log _(ε) cosh x + C
Miscellaneous.
-1/((x + a)^2) 1/(x+a) log _(ε)(x+a) + C
x/((a^2+x^2)^(3/2)) 1/(√(a^2+x^2)) log _(ε)(x + √(a^2+x^2))+C
mp b/((a ± bx)^2) 1/(a ± bx) ± 1/b log _ε (a ± bx) + C)
-(3a^2x)/((a^2 + x^2)^(5/2)) (a^2)/((a^2 + x^2)^(3/2)) x/(√(a^2 + x^2)) + C
a · cos ax sin ax -1/a cos ax + C
-a · sin ax cos ax 1/a sin ax + C
a · sec ^2ax tan ax -1/a log _ε cos ax + C
sin 2x sin ^2 x x/2 - (sin 2x)/4 + C
- sin 2x cos ^2 x x/2 + (sin 2x)/4 + C
n · sin ^(n-1) x · cos x sin ^n x -(cos x)/n sin ^(n-1) x + (n-1)/n ∫ sin ^(n-2) x dx + C
-(cos x)/(sin ^2 x) 1/(sin x) log _ε tan x/2 + C
-(sin 2x)/(sin ^4 x) 1/(sin ^2 x) - cotan x + C
(sin ^2 x - cos ^2 x)/(sin ^2 x · cos ^2 x) 1/(sin x · cos x) log _ε tan x + C
n · sin mx · cos nx + m · sin nx · cos mx sin mx · sin nx 1/2 cos (m - n)x - 1/2 cos (m + n)x + C
2a· sin 2ax sin ^2 ax x/2 - (sin 2ax)/4a + C
-2a· sin 2ax cos ^2 ax x/2 + (sin 2ax)/4a + C
· · ·
End of part 26 of 27
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