Calc 2 Ch 8 Integration Review

Con te Partiro

100

Integrate

int  x e^(4x) dx 

1/4*xe^(4x) - 1/16*e^(4x) + C

200

Integrate

int_0^(pi/4) xcos(2x) dx

[1/2 xsin(2x)+1/4cos(2x)]_0^(pi/4)

= pi/8 - 1/4

300

integrate

int (x+2)e^(2x+1) dx

1/2(x+2)e^(2x+1) - 1/4e^(2x+1) + C

400

Integrate

int x^2 cos(x) dx

x^2sin(x) -2 sin(x) +2x*cos(x) + C

Take the A-Train

100

Integrate

int 1/sqrt(x^2-25) dx

ln|x/5 + sqrt(x^2-25)/5| + C

200

Integrate

int sqrt(x^2-9)/x dx

sqrt(x^2-9) - 3 arccos(3/x) + C

300

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

400

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

Gettin' Triggy with it

100

Integrate

int cos^5(x) sin(x) dx

-cos^6(x)/6 + C

200

Integrate

int cos^2(3x) dx

x/2 + 1/12*sin(6x) + C

300

Integrate

int sec^4(x) dx

1/3*tan^3(x) + tan(x) + C

400

Integrate

int  cot^3(t) / csc(t) dt

-sin(t) - csc(t) + C

To Infinity and Beyond

100

Determine whether the integral converges or diverges.  If it converges evaluate it.

int_2^(oo) 1/x^3 dx

Converges.  1/8

200

Determine whether the integral converges or diverges.  If it converges evaluate it.

int_2^oo 4/x^(1/4) dx

Diverges.

300

Determine whether the integral converges or diverges.  If it converges evaluate it.

int_0^5 10/x dx

Diverges.

400

Determine whether the integral converges or diverges.  If it converges evaluate it.

int_-oo^0 xe^(4x) dx

-1/16

Potpourri

100

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

200

Integrate:

int (x-8)/(x^2-x-6) dx

-ln|x-3| + 2*ln|x+2| + C

300

Use Simpson's Rule with n=4 to approximate:

int_3^4 1/(x-2) dx

Round to 4 decimal places.

0.6933

400

Integrate:

int (4x-2)/[3(x-1)^2] dx

4/3ln|x-1| - 2/[3(x-1)] + C