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H INTEGRAL CALCULUS

QuestionAnswer
Derivative of (u+v-w) u'+v'-w' dx
Derivative of (uv)' u'v+v'u dx
Derivative of (uvw)' u'vw+v'uw+w'uv dx
Derivative of (u/v)' (u'v-v'u)/v² dx
Derivative of [f(u(x))]' f'(u) u'(x) dx
Derivative of (c)' 0
Derivative of (x)' 1 dx
Derivative of (x^n)' nx^(n-1) dx
Derivative of (1/x)' -(1/x²) dx
Derivative of (√x)' (1/2√x) dx
Derivative of (ln x)' (1/x) dx
Derivative of (e^x) e^x dx
Derivative of (a^x) a^(x) ln a dx
Derivative of sin x cos x dx
Derivative of cos x -sin x dx
Derivative of tan x sec^2 x dx
Derivative of cot x - csc² x dx
Derivative of sec x sec x tan x dx
Derivative of csc x -csc x cot x dx
Derivative of sinh x cosh x dx
Derivative of cosh x sinh x dx
Derivative of tanh x sech² x dx
Derivative of sech x -sech x tanh x dx
Derivative of csch x -csch x cot h x dx
Derivative of coth x -csch² x dx
Derivative of arcsin x 1/√(1-x²) dx =>du/√(a²-u²)
Derivative of arccos x -1/√(1-x²) dx =>du/(a²-u²)
Derivative of arctan x 1/(1+x²) dx
Derivative of arcsinh x 1/√(x²+1) dx =>du/√(u²+a²)
Derivative of arccosh x 1/√(x²-1) dx =>du/√(u²-a²)
Derivative of arctanh x 1/(1-x²) dx =>du/(a²-u²)
Integral of 1 dx x
Integral of kf(x) dx k f(x) dx
Integral of [f(x) +/- g(x)] dx separate: Integral of f(x) +/- Integral of g(x)
Integral of x^n dx (x ^(n+1))/n+1 n is not equal to -1
Integral of u^n du generalized power formula: [u^(n+1)]/ n+1
Integral of du/u natural logarithm rule: ln |u| + C
Integral of a^u du exponential function rule: a^u/ln |a| + C
Integral of e^u du exponential function rule: e^u + C
POWERS OF SINE AND COSINE what is: case II use: sin² u = 1/2[(1-cos 2u) cos² u = 1/2[(1+cos 2u) *single even power
POWERS OF SINE AND COSINE what is: case III use: case I sin² u + cos² u = 1 - apply to the other factor w/c have odd power *where power m or n or both are positive odd powers
POWERS OF SINE AND COSINE what is: case IV use: case II sin² u = 1/2[(1-cos 2u); cos² u = 1/2[(1+cos 2u) - apply to both factor *where power m & n are both are positive even powers
POWERS OF SINE AND COSINE what is: case I use: sin² u + cos² u = 1 -singe odd power
POWERS OF TANGENT, COTANGENT,SECANT, COSECANT What is: case I use: tan² u = sec² u - 1; cot² u = csc² u - 1 single odd power, tan/cot is positive integer
POWERS OF TANGENT, COTANGENT,SECANT, COSECANT What is: case II use: sec² u = 1 + tan² u; csc² u = 1 + cot² u -if sec/csc has even power
POWERS OF TANGENT, COTANGENT,SECANT, COSECANT What is: case III use: apply case II to power of sec u /csc u; -power of sec u /csc u is even
POWERS OF TANGENT, COTANGENT,SECANT, COSECANT What is: case IV use: factor sec u tan u/csc u cot u; express tan u/cot u in case I - power of tan u/cot u is odd
Trigonometric identities 1/sin x = ? csc x
Trigonometric identities 1/cos x = ? sec x
Trigonometric identities 1/tan x = ? cot x
Trigonometric identities cos (-x) = ? cos x
Trigonometric identities sin (-x) = ? -sin x
Trigonometric identities quotient: tan (x) = ? sin x/cos x
Trigonometric identities quotient: cot (x) = ? cos x/ sin x
Trigonometric identities double angle: sin (2x) = ? 2sin x cos x
Trigonometric identities quotient: cos (2x) = ? cos² x - sin² x or 2cos² x -1 or 1-2sin² x
Integral of sin x dx -cos x + C
Integral of cos x dx sin x + C
Integral of tan x dx ln|sec x| + C
Integral of cot x dx ln |sin x| + C
Integral of sec x dx ln |sec x + tan x| + C
Integral of csc x dx ln |csc x - cot x| + C
Integral of csc² x dx -cot x + C
Integral of sec ² x dx tan x + C
Integral of csc x tan x dx -csc x + C
Integral of sec x tan x dx sec x + C
Integral of du/[√(a²-u²)] arcsin u/a +C
Integral of du/(a²+u²) 1/a arctan u/a +C
Integral of du/[u√(u²-a²)] 1/a arcsec u/a +C
Integral of du/[√(a²+u²)] arcsinh u/a +C
Integral of du/[√(u²-a²)] arccosh u/a +C
Integral of du/(a²-u²); u<a 1/a arctanh u/a +C
Integral of du/(a²-u²); u>a 1/a arccoth u/a +C
Integral of du/[u√(a²+u²)] -1/a arccsch u/a +C
Trigonometric identities sin A cos B = ? 1/2 [sin (A-B) + sin (A+B)]
Trigonometric identities sin A sin B = ? 1/2 [cos (A-B) - cos (A+B)]
Trigonometric identities cos A cos B = ? 1/2 [cos (A-B) + cos (A+B)]
Integral of sinh x cosh x +C
Integral of cosh x sinh x +C
Integral of tanh x ln |cosh x| +C
Integral of coth x ln |sinh x| +C
Integral of sech x arctanh (sinh x) +C
Integral of csch x ln |tanh u/2 x| +C
Integral of csch² x -coth x +C
Integral of sech² x tanh x +C
Integral of sech x tanh x -sech x +C
Integral of csch x coth x -csch x +C
integration by parts formula? integral of udv = uv - int. vdu
integration by parts: selecting (u)? L - log/ln I - inverse trig A - algebraic E - exponential T - trig
Hyperbolic identities: sinh x = ? (e^x - e^-x)/2
Hyperbolic identities: cosh x = ? (e^x+- e^-x)/2
Hyperbolic identities: tanh x = ? (e^x - e^-x)/(e^x + e^-x)
Hyperbolic identities: cosh² x - sinh² x = ? 1
Hyperbolic identities: 1 - tanh² x = ? sech² x
Hyperbolic identities: coth² x - 1 = ? csch² x
Hyperbolic identities: cosh x - sinh x = ? e^-x
Hyperbolic identities: cosh x + sinh x = ? e^x
Hyperbolic identities: sinh 2x = ? 2sinh x cosh x
Hyperbolic identities: sinh A cosh B = ? 1/2[sinh (A-B) + sinh (A-B)]
Hyperbolic identities: sinh A sinh B = ? 1/2[-cosh (A-B) + cosh (A-B)]
Hyperbolic identities: cosh A cosh B = ? 1/2[cosh (A-B) + cosh (A-B)]
Hyperbolic identities: sinh² x = ? 1/2[cosh 2x - 1]
Hyperbolic identities: cosh² x = ? 1/2[cosh 2x + 1]
Created by: Potpot777
 

 



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