Integral Propterties

,/'cosx dx

sinx + C

,/'sec^2x dx

tanx + C

,/'secxtanx dx

secx + C

Average Value of a Function on an Interval

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

,/' du / (u times the square root of (u^2 - a^2)

(1/a)arcsec(|u| / a) + C

,/' du / (a^2 + u^2)

(1/a)arctan(u/a) + C

,/' x^n dx

(x^(n+1)) / (n + 1) + C, n not equal to -1

Inverse Integral Trig Functions

,/' (dx/ htgtynumbers) OR ,/' (ddhksnumbers/sduhdsnumbers)dx

,/'sinx dx

-cosx + C

,/'csc^2x dx

-cotx + C

,/'cscxcotx dx

-cscx + C

,/'tanx dx

-ln | cosx | + C

,/'cscx dx

-ln | cscx + cotx | + C

The Second Fundamental Theorem of Calculus

d/dx [a,/'x f(t) dt] = f(x)

,/'0dx =

C

If the velocity and acceleration have the same sign, then speed is _______________________

Increasing

A definitive integral will have a ______________ as an answer. The # represents the change in ______________ over the time interval. By the Fundamental Theorem of Calculus, since v(t)=x'(t), the integral will yield x(t2) - x(t1). This is also known as ________________________. The answer can be positive or negative depending on if the particle lands to the left or the right of its initial starting position.

Number; position; displacement

An indefinite integral is an expression for _____________ at time t. Don't forget that you will have a _______________, the value of which can be determined if you know a position value at a particular time.

Position; constant

To find the change in _______________ integrate velocity of the time interval. This is also called ________________________.

Position; displacement

To find the total distance travelled, integrate _______________ ,which is the ____________________________ of velocity

Speed; absolute value

To find a new position of a particle moving along the x-axis, add _________________________ to the integral of ______________ from starting time to new time

Starting position; velocity

An absolute value of a definite integral is a number. This number represents the ________________________________ traveled by the particle over the time interval. The answer should always be ____________________.

Total distance; non-negative

Speed is the absolute value of _________________

Velocity

,/' du / square root of (a^2 - u^2)

arcsin(u/a) + C

,/'kf(x) dx

k ,/' f(x) dx

,/'k dx =

kx + C

,/'secx dx

ln | secx + tanx | + C

,/' cotx dx

ln | sinx | + C