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= + ∫ a dx = ax+ c. ∫ 1 dx = x + c;
Differentiation & Integration formulas Pobierz pdf z
This gives the following formulas (where a ≠ 0), which are valid over any interval where f is continuous (over larger intervals, the constant c must be replaced by a piecewise constant function):
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
Summation of series by integration. Integrals involving ax2 + bx + c; Some generalized results obtained using the fundamental theorems of integrals are remembered as integration formulas in indefinite integration. Dx = e x + c;
You can use the simple formulas for indefinite integral and apply them in your calculations and get the solution easily.
If f(x), g(x) are two functions of a variable x and k is a constant, then (ii) ∫g (x) dx should be taken as the same in both terms. (i) when you find integral ∫g (x) dx then it will not contain an arbitrary constant. If d/dx {φ(x)) = f(x), ∫f(x)dx = φ(x) + c, where c is called the constant of integration or arbitrary constant.
∫(c) = x+c ∫ ( c) = x + c ( where c is a constant) ∫(cx) = cx2 2 +c ∫ ( c x) = c x 2 2 + c ( where c is a constant) ∫(xn) = xn+1 n+1 ∫ ( x n) = x n + 1 n + 1.
A s2 1 area of a triangle: List of basic integration formulas; ∫ (1/x) dx = ln |x| + c ∫ | ( a x + b ) n | d x = sgn ( a x + b ) ( a x + b ) n + 1 a ( n + 1 ) + c {\displaystyle \int \left|(ax+b)^{n}\right|\,dx=\operatorname {sgn}(ax+b){(ax+b)^{n+1} \over a(n+1)}+c}
∫ sec 2 x dx = tan x + c;
Generally, if the function is any trigonometric function, and is its derivative, in all formulas the constant a is assumed to be nonzero, and c denotes the constant of integration. Integrals of exponential and logarithmic functions; Symbols f(x) → integrand f(x)dx → element of integration ∫→ sign of integral φ(x) → anti. If we substitute f (x) = t, then f’ (x) dx = dt.
Let’s discuss some integration formulas by which we can find integral of a function.
∫ x n dx = ((x n+1)/(n+1))+c ; 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: Lim n → ∞ ∑ r = 0 n − 1 f ( r n) ⋅ 1 n = ∫ 0 1 f ( x) d x. ∫ a dx = ax+ c;
Integrals of rational and irrational functions;
The first term consists of the function f (x) and the integral of the function g (x) and the second term constitutes of the whole integration of differentiation of the function f (x) and the integral of the function g. ∫ sec x (tan x) dx = sec x + c. ∫ cos x dx = sin x + c. Express the given series in the form of lim lim n → ∞ ∑ r = 0 n − 1 f ( r n) ⋅ 1 n.
Dx = a x /loga+ c;
∫ e ax cos bx dx = e ax /(a 2 + b 2) (a cos bx + b sin bx) + c. ∫ cos x dx = sin x + c; The integration formula of uv : ∫(ex) = ex +c ∫ ( e x) = e x + c.
The integration by parts formula is calculated as :
Using the fundamental theorems of integrals, there are generalized. ∫ sec2 dx = tan x + c. Integrals with trigonometric functions z sinaxdx= 1 a cosax (63) z sin2 axdx= x 2 sin2ax 4a (64) z sinn axdx= 1 a cosax 2f 1 1 2; Dx = log|x| + c;
∫ 1 dx = x + c.
For a complete list of antiderivative functions, see lists of integrals. ∫(ax) = ax loga +c ∫ ( a x) = a x l o g a + c. For finding sum of an infinite series with the help of definite integration, following formula is used. Dx = x (n + 1) / (n + 1) + c;
The list of integral formulas are given below:
The basic integral formulas are given below: ∫(logax) = 1 xlna +c ∫ ( l o g a x) = 1 x l n a + c. ∫ sec x (tan x) dx = sec x + c; Integration formula in the form e^ax sin bx or e^ax cos bx.
∫a→b f (x) dx = ∫a→c f (x) dx + ∫c→b f (x) dx.
∫(1 x) = ln|x|+c ∫ ( 1 x) = l n | x | + c. ∫a→b f (x) dx = ∫a→b f (t) dt. Integration as inverse operation of differentiation. ∫ 1.dx = x + c;
∫ e x [f(x) + f'(x)].
∫ xn dx = ( (xn+1)/ (n+1))+c ; Thus, it can be seen that the expression of the product rule of integration is divided in two parts; You will very well know the concepts by referring to the antiderivative formulas provided. Integrals involving ax + b;
Here’s the integration formulas list
Contents 1 integrals involving only sine ∫ ( d d x ( f ( x)) ∫ ( g ( x)) d x) d x.
