6 2 homework chapter 7 (sections 7.1 through 7.4)

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7.4E: Exercises for Section 7.4

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• Page ID 70413

• Gilbert Strang & Edwin “Jed” Herman

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Use partial fraction decomposition (or a simpler technique) to express the rational function as a sum or difference of two or more simpler rational expressions.

1) \(\dfrac{1}{(x−3)(x−2)}\)

2) \(\dfrac{x^2+1}{x(x+1)(x+2)}\)

3) \(\dfrac{1}{x^3−x}\)

4) \(\dfrac{3x+1}{x^2}\)

5) \(\dfrac{3x^2}{x^2+1}\) (Hint: Use long division first.)

6) \(\dfrac{2x^4}{x^2−2x}\)

7) \(\dfrac{1}{(x−1)(x^2+1)}\)

8) \(\dfrac{1}{x^2(x−1)}\)

9) \(\dfrac{x}{x^2−4}\)

10) \(\dfrac{1}{x(x−1)(x−2)(x−3)}\)

11) \(\dfrac{1}{x^4−1}=\dfrac{1}{(x+1)(x−1)(x^2+1)}\)

12) \(\dfrac{3x^2}{x^3−1}=\dfrac{3x^2}{(x−1)(x^2+x+1)}\)

13) \(\dfrac{2x}{(x+2)^2}\)

14) \(\dfrac{3x^4+x^3+20x^2+3x+31}{(x+1)(x^2+4)^2}\)

In exercises 15 - 25, use the method of partial fractions to evaluate each of the following integrals.

15) \(\displaystyle ∫\frac{dx}{(x−3)(x−2)}\)

16) \(\displaystyle ∫\frac{3x}{x^2+2x−8}\,dx\)

17) \(\displaystyle ∫\frac{dx}{x^3−x}\)

18) \(\displaystyle ∫\frac{x}{x^2−4}\,dx\)

19) \(\displaystyle ∫\frac{dx}{x(x−1)(x−2)(x−3)}\)

20) \(\displaystyle ∫\frac{2x^2+4x+22}{x^2+2x+10}\,dx\)

21) \(\displaystyle ∫\frac{dx}{x^2−5x+6}\)

22) \(\displaystyle ∫\frac{2−x}{x^2+x}\,dx\)

23) \(\displaystyle ∫\frac{2}{x^2−x−6}\,dx\)

24) \(\displaystyle ∫\frac{dx}{x^3−2x^2−4x+8}\)

25) \(\displaystyle ∫\frac{dx}{x^4−10x^2+9}\)

In exercises 26 - 29, evaluate the integrals with irreducible quadratic factors in the denominators.

26) \(\displaystyle ∫\frac{2}{(x−4)(x^2+2x+6)}\,dx\)

27) \(\displaystyle ∫\frac{x^2}{x^3−x^2+4x−4}\,dx\)

28) \(\displaystyle ∫\frac{x^3+6x^2+3x+6}{x^3+2x^2}\,dx\)

29) \(\displaystyle ∫\frac{x}{(x−1)(x^2+2x+2)^2}\,dx\)

In exercises 30 - 32, use the method of partial fractions to evaluate the integrals.

30) \(\displaystyle ∫\frac{3x+4}{(x^2+4)(3−x)}\,dx\)

31) \(\displaystyle ∫\frac{2}{(x+2)^2(2−x)}\,dx\)

32) \(\displaystyle ∫\frac{3x+4}{x^3−2x−4}\,dx\) (Hint: Use the rational root theorem.)

In exercises 33 - 46, use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals.

33) \(\displaystyle ∫^1_0\frac{e^x}{36−e^{2x}}\,dx\) (Give the exact answer and the decimal equivalent. Round to five decimal places.)

34) \(\displaystyle ∫\frac{e^x\,dx}{e^{2x}−e^x}\,dx\)

35) \(\displaystyle ∫\frac{\sin x\,dx}{1−\cos^2x}\)

36) \(\displaystyle ∫\frac{\sin x}{\cos^2 x+\cos x−6}\,dx\)

37) \(\displaystyle ∫\frac{1−\sqrt{x}}{1+\sqrt{x}}\,dx\)

38) \(\displaystyle ∫\frac{dt}{(e^t−e^{−t})^2}\)

39) \(\displaystyle ∫\frac{1+e^x}{1−e^x}\,dx\)

40) \(\displaystyle ∫\frac{dx}{1+\sqrt{x+1}}\)

41) \(\displaystyle ∫\frac{dx}{\sqrt{x}+\sqrt[4]{x}}\)

42) \(\displaystyle ∫\frac{\cos x}{\sin x(1−\sin x)}\,dx\)

43) \(\displaystyle ∫\frac{e^x}{(e^{2x}−4)^2}\,dx\)

44) \(\displaystyle ∫_1^2\frac{1}{x^2\sqrt{4−x^2}}\,dx\)

45) \(\displaystyle ∫\frac{1}{2+e^{−x}}\,dx\)

46) \(\displaystyle ∫\frac{1}{1+e^x}\,dx\)

In exercises 47 - 48, use the given substitution to convert the integral to an integral of a rational function, then evaluate.

47) \(\displaystyle ∫\frac{1}{t−\sqrt[3]{t}}\,dt; \quad t=x^3\)

48) \(\displaystyle ∫\frac{1}{\sqrt{x}+\sqrt[3]{x}}\,dx; \quad x=u^6\)

49) Graph the curve \(y=\dfrac{x}{1+x}\) over the interval \([0,5]\). Then, find the area of the region bounded by the curve, the \(x\)-axis, and the line \(x=4\).

50) Find the volume of the solid generated when the region bounded by \(y=\dfrac{1}{\sqrt{x(3−x)}}, \,y=0, \,x=1,\) and \(x=2\) is revolved about the \(x\)-axis.

51) The velocity of a particle moving along a line is a function of time given by \(v(t)=\dfrac{88t^2}{t^2+1}.\) Find the distance that the particle has traveled after \(t=5\) sec.

In exercises 52 - 54, solve the initial-value problem for \(x\) as a function of \(t\) .

52) \((t^2−7t+12)\dfrac{dx}{dt}=1,\quad t>4,\, x(5)=0\)

53) \((t+5)\dfrac{dx}{dt}=x^2+1, \quad t>−5,\,x(1)=\tan 1\)

54) \((2t^3−2t^2+t−1)\dfrac{dx}{dt}=3,\quad x(2)=0\)

55) Find the \(x\)-coordinate of the centroid of the area bounded by \(y(x^2−9)=1, \, y=0, \,x=4,\) and \(x=5.\) (Round the answer to two decimal places.)

56) Find the volume generated by revolving the area bounded by \(y=\dfrac{1}{x^3+7x^2+6x},\, x=1,\, x=7\), and \(y=0\) about the \(y\)-axis.

57) Find the area bounded by \(y=\dfrac{x−12}{x^2−8x−20}, \,y=0, \,x=2,\) and \(x=4\). (Round the answer to the nearest hundredth.)

58) Evaluate the integral \(\displaystyle ∫\frac{dx}{x^3+1}.\)

For problems 59 - 62, use the substitutions \(\tan(\frac{x}{2})=t\) , \(dx=\dfrac{2}{1+t^2}\,dt\) ,  \(\sin x=\dfrac{2t}{1+t^2}\) , and \(\cos x=\dfrac{1−t^2}{1+t^2}\) .

59) \(\displaystyle ∫\frac{dx}{3−5\sin x}\)

60) Find the area under the curve \(y=\dfrac{1}{1+\sin x}\) between \(x=0\) and \(x=π.\) (Assume the dimensions are in inches.)

61) Given \(\tan\left(\frac{x}{2}\right)=t,\) derive the formulas \(dx=\dfrac{2}{1+t^2}dt, \,\sin x=\dfrac{2t}{1+t^2}\), and \(\cos x=\dfrac{1−t^2}{1+t^2}.\)

62) Evaluate \(\displaystyle ∫\frac{\sqrt[3]{x−8}}{x}\,dx.\)