Integrals
Evaluate the integral.
∫[(x)/(x4+1)]dx
u=x2 du=2xdx
(1/2)∫[1/(u2+1)]du
(1/2) tan-1u + C
(1/2) tan-1x2 + C
Find the derivative:
f(x)= tanx
f'(x)= sec2x
∫[sinx]dx
cosx + C
Determine whether this series converges or diverges.
Σ(3n/1000)
r=3
diverges by rules of geometric series
What is speed in terms of the velocity function?
s= l v(t) l
Evaluate the integral.
∫[xsinx]dx
u=x du=dx dv=[sinx]dx v=[-cos x ]dx
-xcosx + ∫[cosx]dx
-xcosx + sinx
Find the derivative:
f(x)= cscx
f'(x)= -cscxcotx
∫[csc2x]dx
-cotx + C
Determine whether this series converges or diverges.
Σ(8/9)n
r=8/9
converges by rules of geometric series
what is speed in terms of the position function?
√[(x')2+(y')2]
Evaluate the integral.
∫[10t-3+12t-9+4t3]dt
-5t-2 - (3/2)t-8 + t4 + C
Find the derivative:
f(x)= exe2xe3x
f'(x)= 6e6x
∫[1/(√ (1-x2))]
sin-1x + C
Determine whether this series converges or diverges.
Σ[(3n)/(√n2+4)]
lim as n->∞ = 3
diverges by nth term test
What is the equation for volume according to the Washer Method?
V= pi (the integral from a to b of [R(x)]2)
Evaluate the integral.
∫[1/(x√ x)]dx
∫x(-3/2)dx
(-2/√x)+C
Find the derivative:
f(x)= ex/e2x
f'(x)= -e-x
∫[(cscx)(cotx)]dx
-cscx + C
Determine whether this series converges or diverges.
Σ[((-1)n+1n)/(3n+2)]
lim as n->∞
diverges by the nth term test
If the points (a, f(a)) and (b, f(b)) are on the graph of f(x) the average rate of change of f(x) on the interval [a,b] is
[f(b)-f(a)]/[b-a]
Evaluate the integral.
∫(1+3t)t2dt
∫[t2+3t3]dt
(t3/3)+(3t4/4)+C
Find the derivative:
f(x)= exe4xsinx
f'(x)= e5x(5sinx+cosx)
∫[1/(u(√ u2-a2)]du
Determine whether this series converges or diverges.
Σ[(-1)n/en)]
lim as n->∞
(1/en+1) < (1/en)
What is the equation for volume according to the washer method?
V=pi(the integral from a to b of [R(x)]2-[r(x)]2)