Apr 11 6:05 pm (5 of 15) title: The basic rules of integration, which we will describe below, include the power, constant coefficient (or constant multiplier ), sum, and difference rules. N 6= 1 (2) z 1 x dx = lnjxj (3) z u dv = uv z vdu (4) z e xdx = e (5) z ax dx = 1 lna ax (6) z lnxdx = xlnx x (7) z sinxdx = cosx (8) z cosxdx = sinx (9) z tanxdx = lnjsecxj (10) z secxdx = lnjsecx+tanxj (11) z sec2 xdx = tanx (12) z secxtanxdx = secx (13) z a a2 +x2 dx = tan 1 x a (14) z a a2 x2 dx = 1 2 ln x+a x a (15) z 1 p a2 2x dx = sin 1 x a.
Pin by John Finney on Math Formulas Differentiation and
Table of basic integrals1 (1) z xn dx = 1 n+1 xn+1;
( 2 3)x x dx 2 23 8 5 6 4.
Integration rules and techniques antiderivatives of basic functions power rule (complete) z xn dx= 8 >> < >>: ( ) 3 x dx 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= + Du u c 1 1 n udu cn u n ln du uc u edu e cuu 1 ln adu a cuu a sin cosudu u c cos sinudu u c sec tan2 udu u c csc cot2 uuc csc cot cscuudu uc sec tan secuudu uc 22 1 arctan du u c au a a 22 arcsin du u c au a
Apr 11 6:07 pm (7 of 15).
Apr 11 6:05 pm (4 of 15) title: This page lists some of the most common antiderivatives ∫ cos x dx = sin x + c. Standard integration techniques note that at many schools all but the substitution rule tend to be taught in a calculus ii class.
Answer will be negative (+5) !×!
For indefinite integrals drop the limits of integration. The curve c with equation y = f (x) passes through the point (5, 65). Xn+1 n+ 1 + c; 23 ( ) 2 1 ∫ 5 cosx x dx 3 22 1
The list of basic integral formulas are.
∫ a dx = ax+ c. Integration rules and formulas integral of a function a function ϕ(x) is called a primitive or an antiderivative of a function f(x), if ?'(x) = f(x). Find this pin and more on mathematics by digital study center. ∫ sec 2 x dx = tan x + c.
Dx= ln( 1 x+ a (5) z (x+ a)ndx= (x+ a)n+1.
(5 8 5)x x dx2 2. Basic differentiation rules basic integration formulas derivatives and integrals © houghton mifflin company, inc. Apr 11 6:04 pm (3 of 15) title: If n= 1 exponential functions with base a:
Integration is the basic operation in integral calculus.while differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful.
The cosine of a variable cos (x): Theorem let f(x) be a continuous function on the interval [a,b]. Dx x xx 1 5. ∫ 1 dx = x + c.
Then the collection of all its primitives is called the indefinite integral of f(x) and is denoted by ∫f(x) dx.
Basic trig and exponential examples that use rules from the table of integrals, as well as trig identities. Z ex dx= ex + c if we have base eand a linear function in the exponent, then z eax+b dx= 1 a eax+b + c trigonometric functions z Let f(x) be a function. Apr 11 6:06 pm (6 of 15) title:
∫ sec x (tan x) dx = sec x + c.
( 6 9 4 3)x x x dx32 3 3. We will provide some simple examples to demonstrate how these rules work. 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 If n6= 1 lnjxj+ c;
Apr 11 6:03 pm (2 of 15) title:
Review of difierentiation and integration rules from calculus i and ii for ordinary difierential equations, 3301 general notation: ∫ x n dx = ( (x n+1 )/ (n+1))+c ; Cos ( x) d x = sin ( x) + c. Z ax dx= ax ln(a) + c with base e, this becomes:
Basic integration formulas and the substitution rule 1the second fundamental theorem of integral calculus recall fromthe last lecture the second fundamental theorem ofintegral calculus.
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. The substitution u gx= ( )will convert (( )) ( ) ( ) ( ) b gb( ) a ga ∫∫f g x g x dx f u du= using du g x dx= ′( ). Now use the rules for adding: Basic properties/formulas/rules d (cf x cf x( )) ( ) dx = ′ , is any constant.c (xgxf xgf x( )± =±( ))′ ′′( ) ( ) d (x nxnn) 1 dx = −, n is any number.
= 3:14159¢¢¢ f;g;u;v;f are functions fn(x) usually means [f(x)]n, but f¡1(x) usually means inverse function of f a(x + y) means a times x + y.
