Integration By "Party"
int xln(x) dx
1/2 x^2ln(x)-1/4x^2+C
Evaluate
int_1^2 (x + 2x^2)/x dx
Evaluate
int (4x^3) / sqrt(x^4 + 9) dx
What is
2 sqrt(x^4 + 9) + C
Evaluate
int sin(x)/root3(cos(x)) dx
What is
-3/2 (cos(x))^(2/3) + C
The integral used to find the area bounded by
y = 2^x
, y = 8, and the y-axis. (Must include a sketch of the region)What is
24-7/ln(2)
int (x+2)e^(2x+1) dx
What is
1/2(x + 2)e^(2x + 1) - 1/4 e^(2x + 1) + C
Evaluate
int_e^(e^2) 1/(x lnx) dx
Evaluate
int tan(x) sec^4(x) dx
What is
1/4 sec^4(x) + C or 1/4 tan^4(x) + 1/2 tan^2(x) + C
Evaluate
int sin(x) cos(cos(x)) dx
What is
-sin(cos(x)) + C
The integral used to find the volume obtained by revolving the region bounded by
y = x^2 − 4
andy = 4 − x^2
around the line x = 2? (You must sketch the region.)What is
2pi int_-2^2[(2 - x)((4 - x^2) - (x^2 - 4)]dx
Evaluate
int x^2 cos(x) dx
What is
x^2 sin(x) + 2x cos(x) - 2sin(x) + C
Evaluate
int_0^2 x e^xdx
What is
e^2 + 1
Evaluate
int 1/(xsqrt(lnx)) dx
What is
2 sqrt(lnx) + C
Evaluate
int tan^2(x)sec^4(x) dx
What is
1/5 tan^5(x) + 1/3 tan^3(x) + C
The integral used to compute the volume of the solid obtained by rotating the region enclosed by
y = x^3 and
y = 4x
about y = -1.pi int_0^2((4x + 1)^2 - (x^3 + 1)^2)dx
Evaluate
int e^sqrt(x) dx
What is
2e^sqrt(x) * sqrt(x) - 2 e^sqrt(x) + C
Evaluate
int_(sin(pi/2))^lne cos(e^(x^2))dx
Evaluate
int x sqrt(x - 5) dx
What is
2/5 (x - 5)^(5/2) + 10/3 (x - 5)^(3/2) + C
Evaluate
int (1 + cos(x))/sin(x) dx
What is
ln(csc(x) - cot(x)) + ln(sin(x)) + C
A bucket that weighs 70 lb when filled with water is lifted at a constant rate, by a mechanical winch, from the bottom of a well that is 60 feet deep. The chain that is being used to lift the bucket weighs 0.55 pounds per foot. Now find the amount of work required to lift the bucket from the bottom of the well all the way to the top.
What is 5,190 ft-lbs
Evaluate
int e^x sin(3x) dx
What is
1/10 e^x sin(3x) - 3/10 e^x cos(3x) + C
Evaluate
int_-pi^pi x e^sin(x^2)dx
Evaluate
int sqrt(16 - x^2) dx
What is
1/2 x sqrt(16 - x^2) + 8 arcsin(x/4) + C
Evaluate
int x tan^2(x) dx
What is
x tan(x) - 1/2x^2 + ln(cos(x)) + C
A cylindrical tank with a length of 10 m and a radius of 5 m is on its side and half-full of gasoline. How much work is required to empty the tank through an outlet pipe at the top of the tank? (The density of gasoline is 737 kg/m^3)
What is 20.2 million J