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Numerical Integration
Integrating: Natural Logs & Trig Functions
Integrating: Exponential functions and bases other thane
Integrating: Inverse trig functions and hyperbolic functions
Position, velocity,acceleration, average value
100
∫tanx dx
-ln|cosx| + C
100
∫2^(sinx) cosx dx
2^(sinx) / ln(2) + C
100
∫2 cosh(x)dx
2sinh(x) + C
100
The position of a particle is given by x(t)=t^3-6t^2+9t-2. What is the acceleration of the particle?
a(t) = 6t - 12
200
∫sec2x dx
(1/2) ln|secx + tanx| + C
200
∫e^(-2x)dx
(-1/2) e^(-2x) + C
200
∫1/√(4-x^2 ) dx
arcsin(x/2) + C
200
If the amount of oxygen (in liters) in a chemical process at time t (in seconds) is given by the function v(t)=0.124t^4+3.12t, what is the average amount of oxygen in the process over the first 10 seconds?
263.6 liters
300
∫2 / (x+2) dx
2 ln|x+2| + C
300
∫(e^x + e^-x) / (e^x - e^-x) dx
ln|e^x - e^-x| + C
300
∫cosh(x)/sinh(x) dx
ln|sinh(x)| + C
300
A ball is thrown upward from a point 20 meters above the ground with an initial velocity of 5 meters/sec. The acceleration due to gravity is -9.8 meters/sec^2. What is the function for the position of the ball at time t?
x(t) = -4.9 t^2 + 5 t + 20
400
Use the Trapezoidal Rule to approximate the integral from 0 to 2 of x^3 dx with n = 4.
4.25
400
∫x / (x^2 + 1) dx
(1/2) ln|x^2 + 1| + C
400
∫(6^x - 2^x) dx
(6^x)/ln6 - (2^x)/ln2 + C
400
∫4/(1+9x^2) dx
(4/3) arctan(3x)+C
400
At the instant a traffic light turns green, a car stopped at the intersection accelerates at a constant rate of 6 feet/sec^2. At the same moment, a truck moving at a constant velocity of 36 feet/sec passes the car. When will the car pass the truck?
12 seconds later.
500
Use Simpson's Rule to approximate the integral from 0 to 2 of x^3 dx with n = 4.
4
500
∫(lnx)^2 / x dx
(1/3) ln^3(x) + C
500
∫ln(e^(2x-1))dx
x^2 - x + C
500
∫cosh^2(x-1)sinh(x-1)dx
(1/3) cosh^3(x-1)+C
500
A pumpkin is dropped from the top of a building 100 meters tall. How long will it take the pumpkin to hit the ground? The acceleration due to gravity is -9.8 m/sec^2.
Approximately 4.52 seconds.