Exponent and Logarithm Integrals

Question 1

Sa^xdx

Answer 1

(a^x)/lna+c

Question 2

Se^(5x)dx

Answer 2

U-substitution

Question 3

S(1/x)dx

Answer 3

ln|x|+c

Question 4

Stanxdx

Answer 4

ln|secx|+c

Question 5

Scotxdx

Answer 5

ln|sinx|+c

Question 6

Ssecxdx

Answer 6

ln|secx+tanx|+c

Question 7

Scscxdx

Answer 7

ln|cscx-cotx|+c

Question 8

xyy’=x^2+1

Answer 8

xyy’=x^2+1 -> xydy/dx=x^2+1/x -> Sydy=Sx+x^-12x -> 1/2y^2=1/2x^2+ln|x|+c -> y=sqt(x^2+2ln|x|+2c

Question 9

A bacteria culture starts with 1000 bacteria and after 2 hours, the population is 2500 bacteria. The bacteria culture grows at a rate proportional to its size. Find the expression for the number of bacteria after t hours.

Answer 9

dy/dt=ky -> Sdy/y=Skdt -> ln|y|=kt+c -> |y|=e^kt+c -> y=ce^kt -> 1000=ce^k(0) -> c=1000 -> 2500=1000e^k(2) -> e^2k=5/2 -> ln5/2=2k -> ln2.5/2=k -> y=1000e^ln2.5/2t

Question 10

Ler P(t) represent the number of wolves in a population at time t years when t >= 0. The population P(t) is increasing at a rate directly proportional to 800-P(t), where the constant of proportionality is k. If P(0)=500, find P(t) in terms of t and k.

Answer 10

dP/dt=k(800-P) -> dP/800-P=kdt -> -ln|800-P|=kt+c -> 800-P=e^-kt-c=e^-kte^-c -> 800-P=ce^-kt -> P=800-ce^-kt -> P(0)=800-c=500 -> c=300 -> P(t)=800-300e^-kt

Question 11

Ler P(t) represent the number of wolves in a population at time t years when t >= 0. The population P(t) is increasing at a rate directly proportional to 800-P(t), where the constant of proportionality is k. Find limt->infinityP(t)

Answer 11

limt->infinity800-300e^-kt -> 800-300e^-infinity -> 800-300(0)=800

Question 12

A particle moves along the x-axis with velocity at time t>=0 given by v(t)=-1+e^1-t. Find the total distance traveled by the particle over the time interval 0

Answer 12

d(t)=-t-e^1-t+c -> -1+e^1-t -> t=1 -> d(0), d(1), and d(3) -> 1.853

Question 13

A particle moves along the x-axis so that its velocity v at time t, for 0

Answer 13

1S0ln(t^2-3t+3)dt/1-0=0.554