Integration formulas the following list provides some of the rules for finding integrals and a few of the common antiderivatives of functions. ∫ a dx = ax+ c; We interpreted constant of integration graphically.
Some Important Integral Formulas
This is not the only way to do the algebra, and typically there are many paths to the correct answer.
The formula is the most important reason for including dx in the notation for the definite integral, that is, writing z b a f(x)dx for the integral, rather than simply z b a f(x), as some authors do.
Each formula for the derivative of a specific function corresponds to a formula for the derivative of an elementary function. Integrals of trigonometric functions ∫sin cosxdx x c= − + ∫cos sinxdx x c= + ∫tan ln secxdx x c= + ∫sec ln tan secxdx x x c= + + sin sin cos2 1( ) 2 ∫ xdx x x x c= − + cos sin cos2 1 ( ) 2 ∫ xdx x x x c= + + ∫tan tan2 xdx x x c= − + ∫sec tan2 xdx x c= + integrals of exponential and logarithmic functions ∫ln lnxdx x x x c= − + ( ) 1 1 2 ln ln 1 1 n n x xdx x cn x x n n 7.2.2 some properties of indefinite integral 7.2 integration as an inverse process of differentiation.
The list of basic integral formulas are.
The following table lists integration formulas side by side with the corresponding differentiation formulas. With de nite integrals, the formula becomes z b a udv= u(x)v(x)]b a z b a vdu: Sin 2 2sin cos(x x x)= ( ) ( ), 2 ( ) 1 (( )) cos 1 cos 2xx= + 2 , 2 ( ) 1 (( )) sin 1 cos 2xx= − 2 ex. ∫a→b f (x) dx = ∫a→b f (t) dt.
Basic forms z xndx = 1 n+ 1 xn+1(1) z 1 x dx= lnjxj (2) z udv= uv z vdu (3) z 1 ax+ b dx= 1 a lnjax+ bj (4) integrals of rational functions z 1 (x+ a)2.
For a complete list of integral functions, please see the list of integrals. ∫ (1/x) dx = ln |x| + c 7.2.1 geometrical interpretation of indefinite integral. Linearity af(x)+bg(x)dx = a f(x)dx+b g(x)dx substitution f(w(x))w (x)dx = f(w)dw integration by parts
Below are the integration basic formulas for your ready reference:
We used basic antidifferentiation techniques to find integration rules. Since du/dx = 2x, dx = du/2x, and 6antiderivatives may differ by a constant when we compute an antiderivative, the answer we get can be modified by adding a Integral, we can replace it by du:
∫ x n dx = ((x n+1)/(n+1))+c ;
The integrals of specific functions and structural type formulas. We used basic integration rules to solve problems. Integral of f(x) and is denoted by ∫f(x)dx. ∫ xn dx = ( (xn+1)/ (n+1))+c ;
∫ 1 dx = x + c.
Integration as inverse operation of differentiation. ∫ cos x dx = sin x + c. A constant (the constant of integration) may be added to the right hand side of any of these formulas, but has been suppressed here in. Integration u'vdx(byparts) uv —cos x + c sin x + c in eos + c in + c in + + c in — cot xl + c — amtan — arcsin — arcs inh — arccosh — + c — åsin2r+c tan x — x + c —cot x— x + c du sin x tan sec dr esc dr sin2 x dr dr tan2 x dr cot2 x dr du dy (chain rule) ae —sin cosh x.
2 1 sin ( ) 1 cos(2 )x 2 sin tan cos x x x 1 sec cos x x cos( ) cos( ) x x 22sin ( ) cos ( ) 1xx 2 1 cos ( ) 1 cos(2 )x 2 cos cot sin x x x 1 csc sin x x sin( ) sin( ) x x 22tan ( ) 1 sec ( )x x geometry fomulas:
Multimedia link the following applet shows a. ∫ sec x (tan x) dx = sec x + c. ∫ x n dx = ( (x n+1 )/ (n+1))+c ; ∫ a dx = ax+ c.
∫1/x.dx = log|x| + c.
Generally, if the function is any trigonometric function, and is its derivative, in all formulas the constant a is. ∫ sec2 dx = tan x + c. The following is a list of integrals of exponential functions. If d/dx {φ(x)) = f(x), ∫f(x)dx = φ(x) + c, where c is called the constant of integration or arbitrary constant.
A final property tells one how to change the variable in a definite integral.
Indefinite integrals indefinite integrals are antiderivative functions. For antiderivatives involving both exponential and trigonometric functions, see list of integrals of exponential functions. Ex.∫tan sec35x xdx ( ) ( ) ( ) 35 2 4 24 24 1 1 75 7 5 tan sec tan sec tan sec sec 1sec tan sec 1 sec sec sec x xdx x x x xdx x x x xdx u u du u x x xc = = − =− = = − + ∫∫ ∫ ∫ 5 3 sin cos x x ∫ dx ( ) 11 22 22 Another possibility, for example, is:
Z xn dx = xn+1 n+1 if n 6= −1 d dx (xn.
Some generalised results obtained using the fundamental theorems of integrals are remembered as integration formulas in indefinite integration. Z 2xcos(x2)dx = z cosudu = sinu+c = sin(x2)+ c. ∫ sec 2 x dx = tan x + c; The basic integral formulas are given below:
∫ sec 2 x dx = tan x + c.
∫ a dx = ax+ c. Strip one tangent and one secant out and convert the remaining tangents to secants using tan sec 122xx= −, then use the substitution ux=sec 2. For a complete list of antiderivative functions, see lists of integrals. ∫ 1 dx = x + c;
∫ sec x (tan x) dx = sec x + c;
The list of integral formulas are given below: A s2 1 area of a triangle: ∫ 1 dx = x + c. Symbols f(x) → integrand f(x)dx → element of integration ∫→ sign of integral
We solved general differential equations.
Know more about these integrals class 12 formulas in pdf list. Dx= ln( 1 x+ a (5) z (x+ a)ndx= (x+ a)n+1. Each integral will be dealt with differently. ∫ 1.dx = x + c.
∫ cos x dx = sin x + c.
(this just means we nd the antiderivative using ibp and then plug in the limits of integration the way we do with other de nite integrals.) trigonometric integrals for integrals involving only powers of sine and cosine (both with the same argument): ∫ x n.dx = x (n + 1) / (n + 1)+ c. Table of integrals basic forms (1)!xndx= 1 n+1 xn+1 (2) 1 x!dx=lnx (3)!udv=uv!vdu (4) u(x)v!(x)dx=u(x)v(x)#v(x)u!(x)dx rational functions (5) 1 ax+b!dx= 1 a ln(ax+b) (6) 1 (x+a)2!dx= 1 x+a (7)!(x+a)ndx=(x+a)n a 1+n + x 1+n #$ % &', n!1 (8)!x(x+a)ndx= (x+a)1+n(nx+xa) (n+2)(n+1) (9) dx!1+x2 =tan1x (10) dx!a2+x2 = 1 a tan1(x/a) (11) xdx!a2+x2. 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
∫ cos x dx = sin x + c;
