AB Calculus Final Review

Partial Fraction Integration

100

Integrate the following function with respect to x using partial fractions: 2x/(x²-4)

ln|x²-4|+c

200

Integrate this function with respect to x: (x+2)/(x^2+5x)

(2/5)(ln|x|)+(3/5)(ln|x+5|)+c

300

Integrate the following with respect to x: 1/((x+2)(x+1)(x-3))

(1/5)ln|x+2|+(1/20)ln|x-3|-(1/4)ln|x+1|+c

400

3/[(4x+1)(x+1)] With respect to x, from x=0 to x=2

ln(3) or 1.099

500

3/(x-1)(x+2) dx, from x=2 to x=3

ln(8/5) or 0.47

Integration by Parts I

100

Integrate x(cosx) with respect to x 

x(sinx)+cosx+c

200

Integrate 2x(cos(3x+1)) with respect to x 

(2x/3)*sin(3x+1)+(2cos(3x+1)/9)+c

300

Integrate 4xe^(3x+1) with respect to x 

(4x/3)e^(3x+1) - (4x/9)e^(3x+1) +c 

400

Integrate x^4*lnx, from 1 to e^2 with respect to x 

(2/5)e^10 -(1/25)e^10 +1/25 + c

500

Integrate (3x^2 -2x + 1)(lnx), with respect to x from 0 to e^2

(5/3)(e^6)-(3/2)(e^4)+e^2 or 597.873

Integration by Parts II

100

Integrate (e^x)(cos x) with respect to x

((e^x)(cos x)+(e^x)(sinx))/2 + c

200

Integrate (x^2)(sin x) with respect to x 

5(-(x)^2cosx + 2xsinx + 2cos x) +c 

300

Integrate (x^2)*e^(x+3) with respect to x

(e^(x+3))(x^2-2x+2) +c 

400

Integrate (x^2)(cos x) from x=0 to x=pi, with respect to x

-2pi, or -6.283

500

Integrate x^2*ln^2x with respect to x, from x=0 to x=e^2

(26e^6/27) or 388.487

Integration by Parts III

100

Integrate (x^2)(e^x) with respect to x 

e^x(x^2 - 2x + 2) + c

200

Integrate x^4 * cos x with respect to x 

(x^4)sinx + (4x^3)cosx - (12x^2)sin x -24xcosx +24sinx +c 

300

Integrate x^2(sin 4x) with respect to x 

(1/64)(8x sin(4x) + (2-16x^2)(cos 4x) + c

400

Integrate (x^4)(e^-3x) with respect to x 

-((e^-3x)/81)(27x^4 + 36x^3 + 36x^2 + 24x + 8) + c

500

Integrate (e^x)(sinx) using tabular integration 

(e^x/2)(sinx - cosx) +c 

Volume by Shells

100

Find the volume of the solid obtained by rotating the region bounded by y=x-x^3, y=0, x=0, and x=1 about the y-axis 

4pi/15 or 0.838

200

Find the volume of the solid obtained by rotating the region bounded by the curve y=1/x, the x-axis, over the interval [1,3] about the y-axis. 

4pi, or 12.57

300

Find the volume of the region bounded by the curve y=3x-x^2, and the x-axis from [0,2], about the y-axis

8pi, or 25.13

400

Find the volume of the region bounded by the region y=x^2, the x-axis, over the interval [1,2], about the y-axis

(15pi/2), or 23.56

500

Find the volume when the region bounded by the curves y= -x^2 +7 and y = x^2 + 5 is rotated about the line x=2

(32pi/3) or 33.51