Derivatives and Integral Formulas Flashcards
d/dx(sinax)
acos(ax)
d/dx(cosax)
-asin(ax)
d/dx(tanax)
asec^2(ax)
d/dx(cotx)
acsc^2(ax)
d/dx(secax)
asecaxtanax
d/dx(cscax)
-acscaxcotax
d/dx(e^ax)
ae^ax
d/dx(b^x)
b^xlnb
d/dx(ln|x|)
1/x
Power Rule: d/dx(x^n)
nx^n-1
Product Rule: d/dx(f(x)*g(x))
f’(x)g(x)+g’(x)f(x)
Quotient Rule: d/dx(f(x)/g(x))
f’(x)g(x)+g’(x)f(x)/(g(x))^2
∫ cosaxdx
1/a(sinax)+C
∫ sinaxdx
-1/a(cosax)+C
∫ sec^(2)axdx
1/a(tanax)+C
∫ csc^(2)axdx
-1/a(cotax)+C
∫ secaxtanaxdx
1/a(secax)+C
∫ cscaxcotaxdx
-1/a(cscax)+C
∫ e^axdx
1/ae^ax+C
∫ b^xdx
1/lnb(b^x)+C
∫ (1/x)dx
ln|x|+C
∫ (1/a^2+x^2)
1/a(tan^-1(x/a))+C
∫ (1/xsqrt(x^2-a^2))
1/a(sec^-1|x/a|+C
∫ (1/sqrt(a^2-x^2))
sin-1(x/a)+C
d/dx(sin^-1x)
1/sqrt(1-x^2)
d/dx(cos^-1x)
-1/sqrt(1-x^2)
d/dx(tan^-1x)
1/1+x^2
d/dx(cot^-1x)
-1/1+x^2
d/dx(sec^-1x)
1/xsqrt(x^2-1)
d/dx(csc^-1x)
-1/xsqrt(x^2-1)
rec sec(x)
1/(cosx)
rec csc(x)
1/(sinx)
rec cot(x)
1/(tanx)
rec cos(x)
1/(secx)
rec sin(x)
1/(cscx)
rec tan(x)
1/(cotx)
tan(x)
sinx/cosx
cot(x)
cosx/sinx
sin^2x+cos^2x
1
1+tan^2x
sec^2x
1+cot^2x
csc^2x
sin^2(A)
1/2(1-cos(2A))
cos^2(A)
1/2(1+cos(2A))
sin(2A)
2sin(A)cos(A)
cos(2A)
2cos^2(A)-1,1-2sin^2(A),cos^2(A)-sin^2(A)
∫
integral sign
∫f(x)dx what is f(x)
integrand
∫tan(ax)dx
-1/aln|cos(ax)|+C
∫cot(ax)dx
1/aln|sin(ax)|+C
∫sec(ax)dx
1/aln|sec(ax)+tan(ax)|+C
∫csc(ax)dx
1/aln|csc(ax)+cot(ax)|+C
∫udv
uv-∫vdu
acronym for integration by parts
LIATE, log, inverse trigonometric, algebraic, trignometric, exponential
sin^m(x)cos^n(x) if power of sin is odd
save one sine factor with the dx, pythagorean identity, u-substitution
sin^m(x)cos^n(x) if power of cosine odd
save one cosine factor with the dx, pythagorean identity, u-substitution
sin^m(x)cos^n(x) all powers even
power reducing formula, simplify
sec^m(x)tan^n(x) if the power of secant is even
save a factor of sec^2(x) with dx, Use sec^2(x)=1+tan^2(x), use u-substitution
sec^m(x)tan^n(x) if the power of tangent is odd
save a factor of sec(x)tan(x) with dx, use tan^2(x)=sec^2(x)-1,use u-substitution
∫tan^n(x)dx
(1/n-1)tan^n-1(x)-∫tan^n-2(x)dx
∫sec^n(x)dx
(1/n-1)tan(x)sec^n-2(x)+(n-2/n-1)∫sec^n-2(x)dx
to use trigonometric substitution which 3 expressions must be included
a^2-x^2,x^2-a^2,x^2+a^2
a^2-x^2
x=asin(theta), dx=acos(theta)dx
x^2-a^2
x=asec(theta), dx=asec(theta)tan(theta)dtheta
a^2+x^2
x=atan(theta), dx=asec^2(theta)dtheta