Calculus Flashcards
}udv
uv-}vdu
}ln(x)dx
xln(x)-x+c
}(b,a)u(x)v’(x)dx
u(x)v(x)|(b,a)-}(b,a)v(x)u’(x)
}cos(ax)dx
(1/a)sin(ax)+c
}sin(ax)dx
(-1/a)cos(ax)+c
}sec^2(ax)dx
(1/a)tan(ax)+c
}csc^2(ax)dx
(-1/a)cot(ax)+c
}sec(ax)tan(ax)dx
(1/a)sec(ax)+c
}csc(ax)cot(ax)dx
(-1/a)csc(ax)+c
}e^(ax)dx
(1/a)e^(ax)+c
}b^xdx
(b^x)(1/ln(b))+c
}dx/x
ln|x|+c
d/dx (sin^-1(x/a))
1/(root(a^2-x^2))
For -a<a
d/dx (cos^-1(x/a))
-1/(root(a^2-x^2))
For -a<a
d/dx (sec^-1(x/a))
a/x(root(x^2-a^2))
For |x|>a
d/dx (csc^-1(x/a))
-a/x(root(x^2-a^2))
For |x|>a
d/dx (tan^-1(x/a))
a/a^2+x^2
For -infinity<infinity
d/dx (cot^-1(x/a))
-a/a^2+x^2
For -infinity<infinity
}dx/x(root(x^2-a^2))
(1/a)sec^-1|x/a|+c
For a>0
}dx/root(a^2-x^2))
sin^(x/a)+c
}dx/(a^2+x^2)
(1/a)tan^-1(x/a)+c
}sec(x)dx
ln|sec(x)+tan(x)|+c
}csc(x)dx
-ln|csc(x)+cot(x)|+c
}tan(x)dx
-ln|cos(x)|+c
Or
ln|sec(x)|+c
}cot(x)dx
ln|sin(x)|+c
Even Function:
}(a,-a)f(x)dx
2(}(a,0)f(x)dx)
f(-x)=f(x)
Odd Function:
}(a,-a)f(x)dx
0
f(-x)=-f(x)
Average value over [a,b]
(1/(b-a))(}(b,a)f(x)dx)
}f(g(x))g’(x)dx
}f(u)du+c
}(a,b)f(g(x))g’(x)dx
}(g(b),g(a))f(u)du
Sigma(n,k=1) k^3
(n^2)(n+1)^2/4
Sigma(n,k=1) k^2
(n)(n+1)(2n+1)/6
Sigma(n,k=1) k
(n)(n+1)/2
Sigma(n,k=1) c
cn
Xk of f(Xk) of left Riemann sum
Xk=a+(k-1)(delta x)
Xk of f(Xk) of midpoint Riemann sum
Xk=a+(k-1/2)(delta x)
Xk of f(Xk) of right Riemann sum
Xk=a+k(delta x)
d/dx sin(ax)
(a)cos(ax)+c
d/dx cos(ax)
-(a)sin(ax)+c
d/dx tan(ax)
(a)sec^2(ax)+c
d/dx cot(ax)
-(a)csc^2(ax)+c
d/dx sec(ax)
(a)sec(ax)tan(ax)+c
d/dx csc(ax)
-(a)csc(ax)cot(ax)+c
}b^-xdx
-(b^-x)(1/ln(b))+c
