Integration Rules Table Lesson 4.3 Notes On YouTube

Z cotxdx= ln sinx +c 8. Table of standard integrals 1. Table of integrals engineers usually refer to a table of integrals when performing calculations involving integration.

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If a term in your choice for yp happens to be a

You need to be able to identify the appropriate integration formulas to find the inde.

As per the power rule of integration, if we integrate x raised to the power n, then; 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 Z cosec2 xdx= cotx+c 11. To help us in learning these basic rules, we will recognize an incredible connection between derivatives and integrals.

Z dx x = lnjxj+c 3.

This is the integral of ln (x) multiplied by 1 / 2 and we therefore use rule 2 above to obtain: Apr 11 5:59 pm (1 of 15) title: We now use formula 4.3 in the table of integral formulas to evaluate ∫ ln (x) dx. A table of integrals f(x) r f(x)dx k, any constant kx+c.

∫ (1 / 2) ln (x) dx = (1 / 2) ∫ ln (x) dx.

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 Basic forms z xndx = 1 n +1 xn+1(1) z 1 x dx =ln|x| (2) z udv = uv z vdu (3) z 1 ax + b dx = 1 a ln|ax + b| (4) integrals of rational functions z 1 (x + a)2. ∫(f + g) dx = ∫f dx + ∫g dx. U = u(x) is diﬀerentiable function of x;

If r (x) in (4) is one of the functions in the first column in table 2.1, choose yp in the same line and determine its undetermined coefficients by substituting yp and its derivatives into (4).

If the power of the sine is odd and positive: For the following, the letters a, b, n, and c represent constants. 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. These restrictions are shown in the third column.

We can use this rule,.

For the following, let u and v be functions of x, let n be an integer, and let a, c, and c be constants. General rules derivative rule integration rule sum/di erence rule sum/di erence rule d dx f (x)g =0 r dx r r constant multiple rule constant multiple rule d dx cf (x) =0 r dxc r f product rule integration by parts d dx f (x)g = 0) + r dx r quotient rule (no simple rule corresponds) d dx h f (x) g(x) i = 0 g0 [g(x)]2 Sometimes restrictions need to be placed on the values of some of the variables. Apr 11 6:05 pm (5 of 15) title:

In other words, we will increase the exponent by one and divide.

This page lists some of the most common antiderivatives Apr 11 6:07 pm (7 of 15). Z cosecxdx= ln cosecx cotx +c 13. Apr 11 6:04 pm (3 of 15) title:

Sometimes integration tables only relates two integrals.

This leaﬂet provides such a table. Z tanxdx= ln cosx +c 7. Z dx a 2+x = 1 a tan 1 x a +c 9. Convert the remaining factors to cos( )x (using sin 1 cos22x x.) 1.

Integrals involving sec(x) and tan(x):

When we differentiate we multiply and decrease the exponent by one but with integration, we will do things in reverse. Apr 11 6:03 pm (2 of 15) title: Xn+1 n+ 1 + c; Apr 11 6:05 pm (4 of 15) title:

Fundamental rules ( ) 𝑥

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= + Common integrals ∫k dx k x c= + 1 1 1 nn,1 n x dx x c n+ + ∫ = + ≠− 1 1 ln x dx dx x c x ∫∫− = = + 1 11 ln ax b a dx ax b c + ∫ = ++ ∫ln lnudu u u u c= −+( ) ∫eeuudu c= + ∫cos sinudu u c= + ∫sin cosudu u c− += ∫sec tan2udu u c= + ∫sec tan secu udu u c= + ∫csc cot cscu udu u c−+= ∫csc cot2udu u c− =+ ∫tan lnsecudu u c= + Z secxdx= ln secx+tanx +c 12. Apr 11 6:06 pm (6 of 15) title:

If n= 1 exponential functions with base a:

Z ax dx= ax ln(a) + c with base e, this becomes: Integration rules and techniques antiderivatives of basic functions power rule (complete) z xn dx= 8 >> < >>: 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. Example evaluate i = z dx √ 4x5 +9x4, for x > 0.

Table of basic integrals basic forms (1) z xndx= 1 n+ 1 xn+1;

Entry (15) in the integration tables at the end of the textbook is z dx x2 √ ax +b = − √ ax +b bx − a 2b z dx x √ ax +b. We can rewrite the integral as i = z dx p x4(4x +9) = z dx x2 p (4x +9). Choice rules for the method of undetermined coefficients (a) basic rule. The important rules for integration are:

Z xn dx= xn+1 n+1 +c (n6= 1) 2.

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. Save a du x dx sin( ) ii. 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. This calculus video tutorial explains how to do integration by tables.

∫x n dx = (x n+1 /n+1) + c.

N6= 1 (2) z 1 x dx= lnjxj (3) z udv= uv z vdu (4) z 1 ax+ b dx= 1 a lnjax+ bj integrals of rational functions (5) z 1 (x+ a)2 dx= 1 x+ a (6) z (x+ a)ndx= (x+ a)n+1 n+ 1;n6= 1 (7) z x(x+ a)ndx= (x+ a)n+1((n+ 1)x a) (n+ 1)(n+ 2) (8) z 1 1 + x2 dx= tan 1 x (9) z 1 a2 + x2 dx= 1 a tan 1 x a 1 By this rule the above integration of squared term is justified, i.e.∫x 2 dx. F(x) and g(x) are any continuous functions; If n6= 1 lnjxj+ c;