0 d c dx 1n nd x nx dx sin cos d x x dx sec sec tan d x x x dx 2 tan sec d x x dx cos sin d x x dx csc csc cot d x x x dx 2 cot csc d x x dx lnx xd a a a dx x xd e e dx d d cf x c f x dx dx d d d f x g x f x g x dx dx dx f g f g f g 2 f g fgf g g d f g x f g x g x dx. Integrals with logarithms z lnaxdx xax (42) z lnax x dx= 1 2 (lnax)2 (43) z ln(ax+ b)dx= x+ a ln(ax+ b) x;a6= 0 (44) z ln(x2 + a2) dx = xln(x + a) + 2atan 1 x a 2x (45) x2 a) dx = ) + ln x+ a x a 2 (46) ln ax +bx c dx a 4ac b2 tan 1 2ax+ b p 4ac b2 2x+ b 2a + ln ax2 +bx c (47) z xln(ax+ b)dx= bx 2a 1 4 x2 + 1 2 x2 b2 a2 ln(ax+ b) (48) z xln a2 b2x2 dx= 1 2 x2+ 1 2 x2 a2 b2 ln a2 b2x2 (49). But often, integration formulas are used to find the central points, areas and volumes for the most important things.
calculus cheat sheet I made a sheet much like this when
Use double angle formula for sine and/or half angle formulas to reduce the integral into a form that can be integrated.
In other words, integration is the process of continuous addition and the variable “c” represents the constant of integration.
College of arts and sciences. Logarithms lnxy = lnx+lny, lnxa = alnx, ln1 = 0, elnx = x, lney = y, ax = exlna. ∫ sec2x dx = tan x + c. Trigonometry sin( ) = opp hyp cos( ) = adj hyp csc( ) = hyp opp sec( ) = hyp adj tan( ) = opp adj cot( ) = adj opp csc( ) = 1 sin( ) sec( ) = 1 cos( ) tan( ) = sin( ) cos( ) cot( ) = cos( ) sin( ) sin2( ) + cos2( ) = 1 1 + tan2( ) = sec2( ) 1 + cot2( ) = csc2( ) sin(2 ) = 2sin( )cos( ) cos(2 ) = cos2( ) sin2( ) sin2( ) = 1 cos(2 ) 2 cos2( ) = 1 + cos(2 ) 2 inverse trig.
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Integration formulas z dx = x+c (1) z xn dx = xn+1 n+1 +c (2) z dx x = ln|x|+c (3) z ex dx = ex +c (4) z ax dx = 1 lna ax +c (5) z lnxdx = xlnx−x+c (6) z sinxdx = −cosx+c (7) z cosxdx = sinx+c (8) z tanxdx = −ln|cosx|+c (9) z cotxdx = ln|sinx|+c (10) z secxdx = ln|secx+tanx|+c (11) z cscxdx = −ln |x+cot +c (12) z sec2 xdx = tanx+c (13) z csc2 xdx = −cotx+c (14) z ∫f (x) dx = f (x) + c. Ds=+dxy=fxa££xb 1(dx)2 if ,() dy. Integral formulas of trigonometric functions:
Strip one tangent and one secant out and convert the remaining tangents to secants using tan22xx=.
Cos2θ= cos θ−sin2 θ= 2cos2 θ−1=1− 2sin2 θ ∫ tan x dx = log|sec x| + c. Formula sheet for calculus 2 integrals. ± ² 0 d c dx ± ² 1 n n d x nx dx.
Divide [ab,] into n subintervals of width ∆x and choose * x i from each interval.
Elementary differential and integral calculus formula sheet exponents xa ¢xb = xa+b, ax ¢bx = (ab)x, (xa)b = xab, x0 = 1. Evaluate the integral z eq sin 2q dq. Indefinite integral :∫f x dx f x c( ) = +( ) Common integrals indefinite integral method of substitution ∫ ∫f g x g x dx f u du( ( )) ( ) ( )′ = integration by parts ∫ ∫f x g x dx f x g x g x f x dx( ) ( ) ( ) ( ) ( ) ( )′ ′= − integrals of rational and irrational functions 1 1 n x dx cn x n + = + ∫ + 1 dx x cln x ∫ = + ∫cdx cx c= + 2 2 x ∫xdx c= + 3 2 3 x ∫x dx c= +
Ds=+dxy=fxa££xb 1(dx)2 if ,() dy.
I may keep working on this document as the course goes on, so these notes will not be completely finished until the end of the quarter. Of the equation means integral off (x) with respect to x. Basic properties and formulas if fx and g x are differentiable functions (the derivative exists), c and n are any real numbers, 1. Sin(a+ b) = sin acosb +sin bcos a 8.
1+ tan2 θ= sec2 θ 3.
Integral calculus formula sheet derivative rules: In this example, the key idea is to try integration by parts until you get back to the original integral you wish to evaluate. Therefore when we use our formula for integration by parts we get: Summation formulas 118 appendix c.
Integral calculus formula sheet derivative rules:
Common integrals trig integrals calculus: 1+ cot2 θ= csc2 θ 4. Dx is called the integrating agent. Cos(a− b) = cos acosb +sin asin b 11.
Then ( ) (*) 1 lim i b n a n i f x dx f x x →∞ = ∫ =∑ ∆.
∫ cot x dx = log|sin x| + c. Sin2 θ+ cos2 θ=1 2. Strip 2 secants out and convert rest Cos(a+ b) = cos acosb −sin asin b 10.
Sin(a− b) = sin acosb −sin bcos a 9.
∫ sec x dx = log|sec x + tan x| + c. 0 d c dx nn 1 d xnx dx sin cos d x x dx sec sec tan d x xx dx tan sec2 d x x dx cos sin d x x dx csc csc cot d x xx dx cot csc2 d x x dx d aaaxxln dx d eex x dx dd cf x c f x dx dx Suppose f x( ) is continuous on [ab,]. ∫ cos x dx = sin x + c.
The integration of a function f (x) is given by f (x) and it is represented by:
Common integrals v clx = kx+c idx=lnlxl+c l in c uln (u) —u + c ax +1) on u du = for vann xsecl xdx we have the following : Trigonometry cos0 = sin π 2 = 1, sin0 = cos π 2 =. Integral calculus formula sheet derivative rules: The basic use of integration is to add the slices and make it into a whole thing.
Z 2 1 ln x dx = x ln x 2 1 z x dx x = x ln x 2 1 z 2 1 dx = x ln x 2 1 x 2 1 = 2 ln 2 0 2 + 1 = 2 ln 2 1 example 1.3.
