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. 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. © 2005 paul dawkins derivatives basic properties/formulas/rules d(cf()x)cfx() dx =¢, c is any constant.
2.1.11.12.4 Chapter 4 Indefinite Integrals
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
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
Table of basic integrals basic forms (1) z xndx= 1 n+ 1 xn+1; Udv a b ∫=#uv$% a b −vdu a b ∫ u and v are functions of x. You can see how to use this table of common integrals in the previous section: Z 1 x2 dx = 1 x +c 9.
Product rule [ ]uv uv vu dx d = +′ 4.
Z xndx = xn+1 n+1 +c;n 6= 1 4. Z p xdx = 2 x p 3 +c 10. Z f(x)g0(x)dx = f(x)g(x) z g(x)f0(x)dx integrals of rational and irrational functions 3. ∫sec2(𝑥) 𝑥=tan(𝑥) ∫csc2(𝑥) 𝑥=−cot(𝑥) ∫ 𝑥 sin2(𝑥) =−cot(𝑥) ∫ 𝑥 cos2(𝑥) tan(𝑥) arc trigonometric integrals:
Table of useful integrals, etc.
Z x2 dx = x3 3 +c 8. An+1 0 ∞ ∫ integration by parts: Integrate from x = a to x = b Common integrals inde nite integral method of substitution 1.
Constant multiple rule [ ]cu cu dx d = ′, where c is a constant.
This page lists some of the most common antiderivatives E−ax2dx= 1 2 π a # $% & ’(1 2 0 ∞ ∫ ax xe−2dx= 1 2a 0 ∞ ∫ x2e−ax2dx= 1 4a π a # $% & ’(1 2 0 ∞ ∫ x3e−ax2dx= 1 2a2 0 ∞ ∫ x2ne−ax2dx= 1⋅3⋅5⋅⋅⋅(2n−1) 2n+1an π a $ %& ’ 1 2 0 ∞ ∫ x2n+1e−ax2dx= n! 2an+1 0 ∞ ∫ xne−axdx= n! 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 fundamental theorem of calculus establishes the relationship between indefinite and definite.
Table 2.1, choose yp in the same line and determine its undetermined coefficients by substituting yp and its derivatives into (4). Z 1 x dx = lnjxj+c 5. Z cdx = cx+c 6. ∫ 𝑥 𝑥2+1 =arctan(𝑥) ∫ 𝑥 √1−𝑥2 =arcsin(𝑥) ∫ −1 √1−𝑥2 𝑥=arccos(𝑥) ∫ −1 𝑥2+1 𝑥=arccot(𝑥)
Common integrals polynomials ∫dx x c= + ∫k dx k x c= + 1 1,1 1 x dx x c nnn n = + ≠−+ ∫ + 1 dx x cln x ⌠ = + ⌡ ∫x dx x c−1 = +ln 1 1,1 1 x dx x c nnn n − = +≠−+ ∫ −+ 1 1 dx ax b cln ax b a = ++ + ⌠ ⌡ 1 1 1 p p pq qq q p q q x dx x c x c pq + + = += + ∫ ++ trig functions ∫cos sinudu u c= + ∫sin cosudu u c− += ∫sec tan2udu u c= +
Z f (g(x))g0(x)dx = z f(u)du integration by parts 2. Quotient rule v2 vu uv v u dx d ′− ′ = 0 formulas included in custom cheat sheet. 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+.
Z xdx = x2 2 +c 7.
Common derivatives and integrals provided by the academic center for excellence 1 reviewed june 2008 common derivatives and integrals derivative rules: Z 1 1+x2 dx =. ()0 d c dx =, c is any constant. Sum and difference rule [ ]u v u v dx d ± = ±′ 3.
