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Integrate
int x e^(4x) dx
1/4*xe^(4x) - 1/16*e^(4x) + C
Integrate
int 1/sqrt(x^2-25) dx
ln|x/5 + sqrt(x^2-25)/5| + C
Integrate
int cos^5(x) sin(x) dx
-cos^6(x)/6 + C
Determine whether the integral converges or diverges. If it converges evaluate it.
int_2^(oo) 1/x^3 dx
Converges. 1/8
Use the Trapezoidal Rule to approximate the value of
int_0^2 x^2 dx
with n = 4.
Deltax = 1/2
1/4(0^2 + 2*(1/2)^2 + 2*(1)^2 + 2*(3/2)^2 + 2^2) = 2.75
Integrate
int_0^(pi/4) xcos(2x) dx
[1/2 xsin(2x)+1/4cos(2x)]_0^(pi/4)
= pi/8 - 1/4
Integrate
int sqrt(x^2-9)/x dx
sqrt(x^2-9) - 3 arccos(3/x) + C
Integrate
int cos^2(3x) dx
x/2 + 1/12*sin(6x) + C
Determine whether the integral converges or diverges. If it converges evaluate it.
int_2^oo 4/x^(1/4) dx
Diverges.
Integrate:
int (x-8)/(x^2-x-6) dx
-ln|x-3| + 2*ln|x+2| + C
integrate
int (x+2)e^(2x+1) dx
1/2(x+2)e^(2x+1) - 1/4e^(2x+1) + C
Integrate
int_0^3 sqrt(9-x^2) dx
1/2[x*sqrt(9-x^2) + 9 arcsin(x/3)]
1/2(9*pi/2) = (9pi)/4
Integrate
int sec^4(x) dx
1/3*tan^3(x) + tan(x) + C
Determine whether the integral converges or diverges. If it converges evaluate it.
int_0^5 10/x dx
Diverges.
Use Simpson's Rule with n=4 to approximate:
int_3^4 1/(x-2) dx
Round to 4 decimal places.
0.6933
Integrate
int x^2 cos(x) dx
x^2sin(x) -2 sin(x) +2x*cos(x) + C
Integrate
int x / sqrt(x^2+6x+12) dx
sqrt(x^2 + 6 x + 12) - 3 log((x + 3)/sqrt(3) + sqrt(1/3 (x + 3)^2 + 1)) + C
Integrate
int cot^3(t) / csc(t) dt
-sin(t) - csc(t) + C
Determine whether the integral converges or diverges. If it converges evaluate it.
int_-oo^0 xe^(4x) dx
-1/16
Integrate:
int (4x-2)/[3(x-1)^2] dx
4/3ln|x-1| - 2/[3(x-1)] + C