Calculus

Question 0

}udv

Answer 0

uv-}vdu

Question 1

}ln(x)dx

Answer 1

xln(x)-x+c

Question 2

}(b,a)u(x)v’(x)dx

Answer 2

u(x)v(x)|(b,a)-}(b,a)v(x)u’(x)

Question 3

}cos(ax)dx

Answer 3

(1/a)sin(ax)+c

Question 4

}sin(ax)dx

Answer 4

(-1/a)cos(ax)+c

Question 5

}sec^2(ax)dx

Answer 5

(1/a)tan(ax)+c

Question 6

}csc^2(ax)dx

Answer 6

(-1/a)cot(ax)+c

Question 7

}sec(ax)tan(ax)dx

Answer 7

(1/a)sec(ax)+c

Question 8

}csc(ax)cot(ax)dx

Answer 8

(-1/a)csc(ax)+c

Question 9

}e^(ax)dx

Answer 9

(1/a)e^(ax)+c

Question 10

}b^xdx

Answer 10

(b^x)(1/ln(b))+c

Question 11

}dx/x

Answer 11

ln|x|+c

Question 12

d/dx (sin^-1(x/a))

Answer 12

1/(root(a^2-x^2))

For -a<a

Question 13

d/dx (cos^-1(x/a))

Answer 13

-1/(root(a^2-x^2))

For -a<a

Question 14

d/dx (sec^-1(x/a))

Answer 14

a/x(root(x^2-a^2))

For |x|>a

Question 15

d/dx (csc^-1(x/a))

Answer 15

-a/x(root(x^2-a^2))

For |x|>a

Question 16

d/dx (tan^-1(x/a))

Answer 16

a/a^2+x^2

For -infinity<infinity

Question 17

d/dx (cot^-1(x/a))

Answer 17

-a/a^2+x^2

For -infinity<infinity

Question 18

}dx/x(root(x^2-a^2))

Answer 18

(1/a)sec^-1|x/a|+c

For a>0

Question 19

}dx/root(a^2-x^2))

Answer 19

sin^(x/a)+c

Question 20

}dx/(a^2+x^2)

Answer 20

(1/a)tan^-1(x/a)+c

Question 21

}sec(x)dx

Answer 21

ln|sec(x)+tan(x)|+c

Question 22

}csc(x)dx

Answer 22

-ln|csc(x)+cot(x)|+c

Question 23

}tan(x)dx

Answer 23

-ln|cos(x)|+c

Or

ln|sec(x)|+c

Question 24

}cot(x)dx

Answer 24

ln|sin(x)|+c

Question 25

Even Function:

}(a,-a)f(x)dx

Answer 25

2(}(a,0)f(x)dx)

f(-x)=f(x)

Question 26

Odd Function:

}(a,-a)f(x)dx

Answer 26

0

f(-x)=-f(x)

Question 27

Average value over [a,b]

Answer 27

(1/(b-a))(}(b,a)f(x)dx)

Question 28

}f(g(x))g’(x)dx

Answer 28

}f(u)du+c

Question 29

}(a,b)f(g(x))g’(x)dx

Answer 29

}(g(b),g(a))f(u)du

Question 30

Sigma(n,k=1) k^3

Answer 30

(n^2)(n+1)^2/4

Question 31

Sigma(n,k=1) k^2

Answer 31

(n)(n+1)(2n+1)/6

Question 32

Sigma(n,k=1) k

Answer 32

(n)(n+1)/2

Question 33

Sigma(n,k=1) c

Answer 33

cn

Question 34

Xk of f(Xk) of left Riemann sum

Answer 34

Xk=a+(k-1)(delta x)

Question 35

Xk of f(Xk) of midpoint Riemann sum

Answer 35

Xk=a+(k-1/2)(delta x)

Question 36

Xk of f(Xk) of right Riemann sum

Answer 36

Xk=a+k(delta x)

Question 37

d/dx sin(ax)

Answer 37

(a)cos(ax)+c

Question 38

d/dx cos(ax)

Answer 38

-(a)sin(ax)+c

Question 39

d/dx tan(ax)

Answer 39

(a)sec^2(ax)+c

Question 40

d/dx cot(ax)

Answer 40

-(a)csc^2(ax)+c

Question 41

d/dx sec(ax)

Answer 41

(a)sec(ax)tan(ax)+c

Question 42

d/dx csc(ax)

Answer 42

-(a)csc(ax)cot(ax)+c

Question 43

}b^-xdx

Answer 43

-(b^-x)(1/ln(b))+c