Derivatives Basic Properties/Formulas/Rules
d (cf (x))= cf ′(x), c is any constant. dx
d (xn )= nxn−1 , n is any number. dx
′ ′ ′ (f(x)±g(x))=f (x)± g(x)
d (c)= 0, c is any constant. dx
f ′ (f g) = f′g f g+′ –(ProductRule) g =
f′g− f g′ g2
′
d (f(g(x)))= f′(g(x))g′(x) (ChainRule)
–(QuotientRule)
Common Derivatives and Integrals
d(lng(x))=g′(x)
d (cx)=c
dx dx dx
d (secx)=secxtanx d (cscx)= −cscxcotx d (cotx)= −csc2 x dx dx dx
dx
(()) () d egx =g′(x)egx
dx g(x)
dx
Common Derivatives
Polynomials
d (xn )=nxn−1
dx dx dx dx dx
d (cxn )=ncxn−1 d (sinx)=cosx d (cosx)= −sinx d (tanx)=sec2 x
d (c)=0 d (x)=1 Trig Functions
Inverse Trig Functions d(sin−1x)= 1
d(tan−1x)= 1 dx 1+ x2
d (cot−1 x)= − 1 xx2−1 dx 1+x2
dx
d(cos−1x)=− 1
d (sec−1 x)= dx
1− x2 1
dx
d (csc−1 x)= −
1− x2 1
xx2−1 dx Exponential/Logarithm Functions
d (ax)=ax ln(a) d (ex)=ex dx dx
d(ln(x))=1, x 0 > d(lnx)=1, x 0 ≠ d(loga(x))= 1 , x 0 > dx x dx x dx xlna
Hyperbolic Trig Functions
d (sinh x)= cosh x d (cosh x)= sinh x d (tanh x)= sech2 x dx dx dx
d (sech x)= −sech xtanh x d (csch x)= −csch xcoth x d (coth x)= −csch2 x dx dx dx
Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes. © 2005 Paul Dawkins
Common Integrals
Polynomials ∫dx=x c+
∫kdx=kx c+ ∫x−1 dx = ln x c +
1
∫xndx=n+1xn+1 c,n +1 ≠−
Common Derivatives and Integrals
Integrals Basic Properties/Formulas/Rules
cf (x)dx=c f (x)dx,cisaconstant. f (x)±g(x)dx= f (x)dx ±g(x)dx ∫∫∫∫∫
∫b f (x)dx=F(x)b F(b=) F(a) wher−e F(x)=∫f (x)dx aa
cf (x)dx=c f (x)dx,cisaconstant. f (x)±g(x)dx= f (x)dx ±g(x)dx ∫b ∫b ∫b ∫b ∫b
aaaaa a f(x)dx=0 a f(x)dx=−b f(x)dx
∫a ∫b∫a ∫b f (x)dx=∫c f (x)dx ∫b f (x)dx+ ∫bcdx=c(b a)−
aaca If f(x)≥0ona≤x≤bthen∫abf(x)dx≥0
If f(x)≥g(x)ona≤x≤bthen∫bf(x)dx≥∫bg(x)dx aa
⌠ 1 dx = ln x
⌡x −n+1
c +
⌠ dx=lnaxbc++
∫x−ndx = 1 x−n+1 c, n +1
≠
+
11
p 1p+1 qp+q ∫xqdx=p xq c +x=q c
⌡ax+b a Trig Functions
∫cosudu=sinu c + ∫secutanudu =secu c ∫ tan u du = ln sec u c
∫sinud−u= c+osu c + ∫cscucotud−u = c+scu
q +1 c
p+q ∫sec2udu=tanu c +
+ ∫ cot u du = ln sin u
secudu=lnsecu t+anu c+ sec udu=2 secutanu lnsec+u tanu+ c +
∫∫31() cscudu=lncscu c−otu c+ csc udu=− cscuco+tu lncs−cu cot+u c
∫∫31() 2
Exponential/Logarithm Functions
∫eu du=eu c+ ∫au du= au c +
∫e sin(bu)du = eau (asin(bu) bcos(−bu)) c 22
+ ∫ue du =(−u 1)e+ c
+ ⌠ 1 du = ln lnu c +
a+b
∫eau cos(bu)du = eau (acos(bu) bsin(b+u)) c
lna
∫csc2 ud−u = c+otu c c +
∫lnudu=uln(u) u c− + au uu
a2 +b2
Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes.
⌡ulnu
© 2005 Paul Dawkins
Inverse Trig Functions ⌠1u
Common Derivatives and Integrals
−1 ∫−1 −1 2
⌡ a2−u2 du=sin a c +sin udu=usin u 1+u −c +
⌠1 1−1u ∫−1 −1 1(2) a2+u2du=atanac t+anudu=utanu 2ln−1u+c+
⌡ ⌠11u
−1 ∫−1 −1 2 ⌡uu2−a2du=asecac cos+udu=ucosu 1−u−c+
Hyperbolic Trig Functions
∫sinhudu=coshu c +∫sechutanhu−du= s+echu c ∫sech2 udu=tanhu c +
∫coshudu=sinhu c +∫cschucothu−du= c+schu c ∫csch2u−du= c+othu c ∫tanhudu=ln(coshu) c ∫sechu+du=tan−1 sinhu+c
Miscellaneous
⌠ 1 du = 1 ln u + a + c ⌠ 1
∫ ∫
du = 1 ln u − a + c 2a u+a
⌡a2 −u2 2a u−a ⌡u2 −a2
22u22a2 22 a+udu= a u+ +lnu a+ u+c+
22 22u22a2 22
u−adu= u a− −lnu u+ a−c+ 22
u a2 u ∫a2−u2du= a2 u2−+sin−1c+
22a
u−a a2 a−u
∫2 2−1 2au−u du= 2 2au u 2−cos+ a +c
Standard Integration Techniques
Note that all but the first one of these tend to be taught in a Calculus II class.
u Substitution
Given ∫ab f (g (x))g′(x)dx then the substitution u = g (x) will convert this into the
integral, ∫b f (g(x))g′(x)dx=∫g(b) f (u)du. a g(a)
Integration by Parts
The standard formulas for integration by parts are,
∫udv=uv−∫vdu ∫budv=uvba −∫bvdu aa
Choose u and dv and then compute du by differentiating u and compute v by using the factthat v=∫dv.
Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes. © 2005 Paul Dawkins
Common Derivatives and Integrals
Trig Substitutions
If the integral contains the following root use the given substitution and formula.
a2 −b2x2 ⇒
b2x2 −a2 ⇒
a2 +b2x2 ⇒ Partial Fractions
x=asinθ b
x=asecθ b
x=atanθ b
and cos2θ−=1 sin2θ and tan2θ=sec2θ−1 and sec2θ+=1 tan2θ
If integrating ⌠ P(x)dx where the degree (largest exponent) of P(x) is smaller than the ⌡ Q(x)
degree of Q(x) then factor the denominator as completely as possible and find the partial
fraction decomposition of the rational expression. Integrate the partial fraction decomposition (P.F.D.). For each factor in the denominator we get term(s) in the decomposition according to the following table.
Factor in Q(x) ax+b
2
∫sinn xcosm xdx
1. If n is odd. Strip one sine out and convert the remaining sines to cosines using sin2 x =1 −cos2 x , then use the substitution u = cos x
2. If m is odd. Strip one cosine out and convert the remaining cosines to sines using cos2 x =1 s−in2 x , then use the substitution u = sin x
3. If n and m are both odd. Use either 1. or 2.
4. If n and m are both even. Use double angle formula for sine and/or half angle
formulas to reduce the integral into a form that can be integrated.
∫tann xsecm xdx
1. If n is odd. Strip one tangent and one secant out and convert the remaining tangents to secants using tan2 x = sec2 x 1, the−n use the substitution u = sec x
2. If m is even. Strip two secants out and convert the remaining secants to tangents using sec2 x =1 +tan2 x , then use the substitution u = tan x
3. If n is odd and m is even. Use either 1. or 2.
4. If n is even and m is odd. Each integral will be dealt with differently.
Term in P.F.D
A ax+b
Factor in Q(x) (ax+b)k
Term in P.F.D
AAA
1 + 2 2++ k k
ax +bx+c
Products and (some) Quotients of Trig Functions
ax+b (ax+b) (ax+b) Ax+B Ax+B
Ax+B2 k11++kk 22k
ax2 +bx+c (ax +bx+c) ax +bx+c (ax +bx+c)
33 622
Convert Example : cos x = (cos x) (1= sin −x)
Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes. © 2005 Paul Dawkins