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INTEGRATION TABLE (INTEGRALS) Notation: f(x) and g(x) are any continuous functions; u = u(x) is differentiable function of x; du = du dx dx = u0 dx; c, n, and a > 0 are constants (1) Z (f(x)+g(x))dx = Z f(x)dx+ Z g(x)dx (2) Z cf(x)dx = c Z f(x)dx (3) Z un du = un+1 n+1 +C, n 6= −1 (a) Z 1 u du = Z du u = ln|u|+C (b) Z 1 √ u du = Z du √ u = 2 √ u+C (c) Z du = u+C (4) Z e udu = e +C (5) Z

An indefinite integral computes the family of functions that are the antiderivative. A definite integral is used to compute the area under the curve These are some of the most frequently encountered rules for differentiation and integration. For the following, let u and v be functions of x, let n be an integer, and let a, c, and C be constants.

integral, (( )) ( ) ( ) ( ) b gb( ) a ga ∫∫f g x g x dx f u du′ = . Integration by Parts The standard formulas for integration by parts are, bb b aa a ∫∫ ∫ ∫udv uv vdu=−= udv uv vdu− Choose uand then compute and dv du by differentiating u and compute v by using the fact that v dv=∫.

4 e4x + c. 4. Integrating trigonometric functions. We know that the derivative of sin x is cos x. Thus. ∫ cosxdx ...

Sometimes we can work out an integral, because we know a matching derivative. ... From the table above it is listed as being −cos(x) + C. It is written as:.

Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative ... These tables, which contain mainly integrals of elementary functions, ...

derivative_integrals.qxd Author: ewedzikowski Created Date: 10/29/2004 9:36:46 AM ...

Table 2.1, choose Yp in the same line and determine its undetermined coefficients by substituting Yp and its derivatives into (4). (b) Modification Rule. If a term in your choice for Yp happens to be a solution of the homogeneous ODE corresponding to (4), multiply this term by x (or by x 2 if this solution corresponds to a double root of the

integral, (())() () bgb( ) aga òòfgxg¢ xdx= fudu . Integration by Parts The standard formulas for integration by parts are, bbb aaa òudv=uv-vduòòudv=-uvvdu Choose u and dv and then compute du by differentiating u and compute v by using the fact that v= …

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Derivations & Integrals Equations. [INTEGRAL TABLES] --- Algebric functions of the forms: a 2 + x 2 a 2 − x 2 x 2 − a 2

Calculus: differentials and integrals, partial derivatives and differential equations. An introduction for physics students. Analytical and numerical differentiation and integration. Partial derivatives. The chain rule. Mechanics with animations and video film clips.

Integral and derivative Table. In this table, a is a constant, while u, v, w are functions. The derivatives are expressed as.

Table 2.1, choose Yp in the same line and determine its undetermined coefficients by substituting Yp and its derivatives into (4). (b) Modification Rule. If a term in your choice for Yp happens to be a solution of the homogeneous ODE corresponding to (4), multiply this term by x (or by x 2 if this solution corresponds to a double root of the

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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

CALCULUS TRIGONOMETRIC DERIVATIVES AND INTEGRALS STRATEGY FOR EVALUATING R sinm(x)cosn(x)dx (a) If the power n of cosine is odd (n =2k +1), save one cosine factor and use cos2(x)=1sin2(x)to express the rest of the factors in terms of sine:

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+x"a) (n+2)(n+1) (9) dx!1+x2 =tan"1x (10) dx!a2+x2 = 1 a tan"1(x/a) (11) xdx!a2+x2 = 1 2 ln(a2+x2) (12) x2dx!a2+x2 =x"atan"1(x/a)

When finding the derivatives of trigonometric functions, non-trigonometric derivative rules are often incorporated, as well as trigonometric derivative ...

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 ...

Table of Integrals∗ 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 dx= ln(1 x+ a (5) Z (x+ a)ndx= (x+ a)n+1 n+ 1;n6= 1 (6) Z x(x+ a)ndx= (x+ a)n+1((n+ 1)x a) (n+ 1)(n+ 2) (7) Z 1 1 + x2 dx= tan 1 x (8) Z 1 a2 + x2 dx= 1 a tan 1 x a (9) Z x a 2+ x dx= 1 2 lnja2 + x2j (10) Z x2 a 2+ x dx= x atan 1 x a (11) Z x3 a 2+ x

Common Derivatives and Integrals ... The standard formulas for integration by parts are, ... decomposition according to the following table. Factor in ( ).

Standard Integration Techniques Note that at many schools all but the Substitution Rule tend to be taught in a Calculus II class. u ′Substitution : The substitution u gx= ( )will convert (( )) ( ) ( ) ( ) b gb( ) a ga ∫∫f g x g x dx f u du= using du g x dx= ′( ). For indefinite integrals drop the limits of integration. Ex. 23 ( ) 2 1 ...

Solve any integral on-line with the Wolfram Integrator (External Link) ... files posted on the web site http://integral-table.com and its derivative pages.

Integration Formulas. ∫ dx = x + C. (1). ∫ xn dx = xn+1 n + 1. + C. (2). ∫ dx x. = ln |x| + C. (3). ∫ ex dx = ex + C. (4). ∫ ax dx = 1 ln a ax + C.

Integrals tables of the form. Derivation Basic Rules. Derivatives - Algebric Functions. Derivatives - Trigonometric Functions. Partial Derivation.