Solved: Arc length of the natural logarithm Consider the

What kinds of functions can be integrated using partial fraction decomposition?

Give an example of each of the following. a. A simple linear factor b. A repeated linear factor c. A simple irreducible quadratic factor d. A repeated irreducible quadratic factor

What term(s) should appear in the partial fraction decomposition of a proper rational function with each of the following? a. A factor of x - 3 in the denominator b. A factor of 1x - 423 in the denominator c. A factor of x2 + 2x + 6 in the denominator

What is the first step in integrating x2 + 2x - 3 x + 1 ?

512. Setting up partial fraction decomposition Give the partial fraction decomposition for the following functions. 2 x2 - 2x - 8

512. Setting up partial fraction decomposition Give the partial fraction decomposition for the following functions. x - 9 x2 - 3x - 18

512. Setting up partial fraction decomposition Give the partial fraction decomposition for the following functions. 5x - 7 x2 - 3x + 2

512. Setting up partial fraction decomposition Give the partial fraction decomposition for the following functions. 11x - 10 x2 - x

512. Setting up partial fraction decomposition Give the partial fraction decomposition for the following functions. x2 x3 - 16x , x _ 0

512. Setting up partial fraction decomposition Give the partial fraction decomposition for the following functions. x2 - 3x x3 - 3x2 - 4x , x _ 0

512. Setting up partial fraction decomposition Give the partial fraction decomposition for the following functions. x + 2 x3 - 3x2 + 2x

512. Setting up partial fraction decomposition Give the partial fraction decomposition for the following functions. x2 - 4x + 11 1x - 321x - 121x + 12

1326. Simple linear factors Evaluate the following integrals. L 3 1x - 121x + 22 dx

1326. Simple linear factors Evaluate the following integrals. L 8 1x - 221x + 62 dx

1326. Simple linear factors Evaluate the following integrals. L 6 x2 - 1 dx

1326. Simple linear factors Evaluate the following integrals. L 1 0 dt t2 - 9

1326. Simple linear factors Evaluate the following integrals. L 2 -1 5x x2 - x - 6 dx

1326. Simple linear factors Evaluate the following integrals. L 21x2 x3 - x2 - 12x dx

1326. Simple linear factors Evaluate the following integrals. L 10x x2 - 2x - 24 dx

1326. Simple linear factors Evaluate the following integrals. L y + 1 y3 + 3y2 - 18y dy

1326. Simple linear factors Evaluate the following integrals. L 6x2 x4 - 5x2 + 4 dx

1326. Simple linear factors Evaluate the following integrals. L 4x - 2 x3 - x dx

1326. Simple linear factors Evaluate the following integrals. L x2 + 12x - 4 x3 - 4x dx

1326. Simple linear factors Evaluate the following integrals. L z2 + 20z - 15 z3 + 4z2 - 5z dz

1326. Simple linear factors Evaluate the following integrals. L dx x4 - 10x2 + 9

1326. Simple linear factors Evaluate the following integrals. L 5 0 2 x2 - 4x - 32 dx

2737. Repeated linear factors Evaluate the following integrals. L 81 x3 - 9x2 dx

2737. Repeated linear factors Evaluate the following integrals. L 16x2 1x - 621x + 222 dx

2737. Repeated linear factors Evaluate the following integrals. L 1 -1 x 1x + 322 dx

2737. Repeated linear factors Evaluate the following integrals. L dx x3 - 2x2 - 4x + 8

2737. Repeated linear factors Evaluate the following integrals. L 2 x3 + x2 dx

2737. Repeated linear factors Evaluate the following integrals. L 2 1 2 t31t + 12 dt

2737. Repeated linear factors Evaluate the following integrals. L x - 5 x21x + 12 dx

2737. Repeated linear factors Evaluate the following integrals. L x2 1x - 223 dx

2737. Repeated linear factors Evaluate the following integrals. L x2 - x 1x - 221x - 322 dx

2737. Repeated linear factors Evaluate the following integrals. L 12y - 8 y4 - 2y2 + 1 dy

2737. Repeated linear factors Evaluate the following integrals. L x2 - 4 x3 - 2x2 + x dx

3841. Setting up partial fraction decompositions Give the appropriate form of the partial fraction decomposition for the following functions. 2 x1x2 - 6x + 92

3841. Setting up partial fraction decompositions Give the appropriate form of the partial fraction decomposition for the following functions. 20x 1x - 1221x2 + 12

3841. Setting up partial fraction decompositions Give the appropriate form of the partial fraction decomposition for the following functions. x2 x31x2 + 12

3841. Setting up partial fraction decompositions Give the appropriate form of the partial fraction decomposition for the following functions. 2x2 + 3 1x2 - 8x + 1621x2 + 3x + 42

4250. Simple irreducible quadratic factors Evaluate the following integrals. L 81x2 + 42 x1x2 + 82 dx

4250. Simple irreducible quadratic factors Evaluate the following integrals. L x2 + x + 2 1x + 121x2 + 12 dx

4250. Simple irreducible quadratic factors Evaluate the following integrals. L x2 + 3x + 2 x1x2 + 2x + 22 dx

4250. Simple irreducible quadratic factors Evaluate the following integrals. L 2x2 + 5x + 5 1x + 121x2 + 2x + 22 dx

4250. Simple irreducible quadratic factors Evaluate the following integrals. L z + 1 z1z2 + 42 dz

4250. Simple irreducible quadratic factors Evaluate the following integrals. L 20x 1x - 121x2 + 4x + 52 dx

4250. Simple irreducible quadratic factors Evaluate the following integrals. L 2x + 1 x2 + 4 dx

4250. Simple irreducible quadratic factors Evaluate the following integrals. L x2 x3 - x2 + 4x - 4 dx

4250. Simple irreducible quadratic factors Evaluate the following integrals. L dy 1y2 + 121y2 + 22

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. To evaluate L 4x6 x4 + 3x2 dx, the first step is to find the partial fraction decomposition of the integrand. b. The easiest way to evaluate L 6x + 1 3x2 + x dx is with a partial fraction decomposition of the integrand. c. The rational function f 1x2 = 1 x2 - 13x + 42 has an irreducible quadratic denominator. d. The rational function f 1x2 = 1 x2 - 13x + 43 has an irreducible quadratic denominator.

5255. Areas of regions Find the area of the following regions. The region bounded by the curve y = x>11 + x2, the x-axis, and the line x = 4

5255. Areas of regions Find the area of the following regions. The region bounded by the curve y = 10>1x2 - 2x - 242, the x-axis, and the lines x = -2 and x = 2

5255. Areas of regions Find the area of the following regions. The region bounded by the curves y = 1>x, y = x>13x + 42, and the line x = 10

5255. Areas of regions Find the area of the following regions. The region bounded by the curve y = x2 - 4x - 4 x2 - 4x - 5 and the x-axis

5661. Volumes of solids Find the volume of the following solids. The region bounded by y = 1>1x + 12, y = 0, x = 0, and x = 2 is revolved about the y-axis.

5661. Volumes of solids Find the volume of the following solids. The region bounded by y = x>1x + 12, the x-axis, and x = 4 is revolved about the x-axis.

5661. Volumes of solids Find the volume of the following solids. The region bounded by y = 11 - x22-1>2 and y = 4 is revolved about the x-axis.

5661. Volumes of solids Find the volume of the following solids. The region bounded by y = 1 2x13 - x2 , y = 0, x = 1, and x = 2 is revolved about the x-axis.

5661. Volumes of solids Find the volume of the following solids. The region bounded by y = 1 24 - x2 , y = 0, x = -1, and x = 1 is revolved about the x-axis.

5661. Volumes of solids Find the volume of the following solids. The region bounded by y = 1>1x + 22, y = 0, x = 0, and x = 3 is revolved about the line x = -1.

Whats wrong? Why are there no constants A and B satisfying x2 1x - 421x + 52 = A x - 4 + B x + 5 ?

6374. Preliminary steps The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. L dx 1 + ex

6374. Preliminary steps The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. L x4 + 1 x3 + 9x dx

6374. Preliminary steps The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. L 3x2 + 4x - 6 x2 - 3x + 2 dx

6374. Preliminary steps The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. L 2z3 + z2 - 6z + 7 z2 + z - 6 dz

6374. Preliminary steps The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. L dt 2 + et

6374. Preliminary steps The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. L dx ex + e2x

6374. Preliminary steps The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. L sec t 1 + sin t dt

6374. Preliminary steps The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. L 2ex + 1 dx 1Hint: Let u = 2ex + 1.2

6374. Preliminary steps The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. L ex 1ex - 121ex + 22 dx

6374. Preliminary steps The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. L cos u 1sin3 u - 4 sin u2 du

6374. Preliminary steps The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. L dx 1ex + e-x22

6374. Preliminary steps The following integrals require a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate these integrals. L dy y11a - 1y2 , for a 7 0. 1Hint: Let u = 1y.2 T

Another form of L sec x dx. a. Verify the identity sec x = cos x 1 - sin2 x . b. Use the identity in part (a) to verify that L sec x dx = 1 2 ln ` 1 + sin x 1 - sin x ` + C. (Source: The College Mathematics Journal 32, 5, Nov 2001)

7681. Fractional powers Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. L dx x - 23 x ; x = u3

7681. Fractional powers Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. L dx 24 x + 2 + 1 ; x + 2 = u4

7681. Fractional powers Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. L dx x11 + 2x ; 1 + 2x = u2

7681. Fractional powers Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. L dx 1x + 23 x ; x = u6

7681. Fractional powers Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. L dx x - 24 x ; x = u4

7681. Fractional powers Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. L dx 21 + 1x ; x = 1u2 - 122

Arc length of the natural logarithm Consider the curve y = ln x. a. Find the length of the curve from x = 1 to x = a and call it L1a2. (Hint: The change of variables u = 2x2 + 1 allows evaluation by partial fractions.) b. Graph L1a2. c. As a increases, L1a2 increases as what power of a?

8386. Repeated quadratic factors Refer to the summary box (Partial Fraction Decompositions) and evaluate the following integrals. L 2 x1x2 + 122 dx

8386. Repeated quadratic factors Refer to the summary box (Partial Fraction Decompositions) and evaluate the following integrals. L dx 1x + 121x2 + 2x + 222

8386. Repeated quadratic factors Refer to the summary box (Partial Fraction Decompositions) and evaluate the following integrals. L x 1x - 121x2 + 2x + 222 dx

8386. Repeated quadratic factors Refer to the summary box (Partial Fraction Decompositions) and evaluate the following integrals. L x3 + 1 x1x2 + x + 122 dx

Two methods Evaluate L dx x2 - 1 , for x 7 1, in two ways: using partial fractions and a trigonometric substitution. Reconcile your two answers.

8894. Rational functions of trigonometric functions An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution u = tan 1x>22 or equivalently x = 2 tan-1 u. The following relations are used in making this change of variables. A: dx = 2 1 + u2 du B: sin x = 2u 1 + u2 C: cos x = 1 - u2 1 + u2 Verify relation A by differentiating x = 2 tan-1 u. Verify relations B and C using a right-triangle diagram and the double-angle formulas sin x = 2 sin x 2 cos x 2 and cos x = 2 cos2 x 2 - 1.

8894. Rational functions of trigonometric functions An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution u = tan 1x>22 or equivalently x = 2 tan-1 u. The following relations are used in making this change of variables. A: dx = 2 1 + u2 du B: sin x = 2u 1 + u2 C: cos x = 1 - u2 1 + u2 Evaluate L dx 1 + sin x .

8894. Rational functions of trigonometric functions An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution u = tan 1x>22 or equivalently x = 2 tan-1 u. The following relations are used in making this change of variables. A: dx = 2 1 + u2 du B: sin x = 2u 1 + u2 C: cos x = 1 - u2 1 + u2 Evaluate L dx 2 + cos x .

8894. Rational functions of trigonometric functions An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution u = tan 1x>22 or equivalently x = 2 tan-1 u. The following relations are used in making this change of variables. A: dx = 2 1 + u2 du B: sin x = 2u 1 + u2 C: cos x = 1 - u2 1 + u2 Evaluate L dx 1 - cos x .

8894. Rational functions of trigonometric functions An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution u = tan 1x>22 or equivalently x = 2 tan-1 u. The following relations are used in making this change of variables. A: dx = 2 1 + u2 du B: sin x = 2u 1 + u2 C: cos x = 1 - u2 1 + u2 Evaluate L dx 1 + sin x + cos x .

8894. Rational functions of trigonometric functions An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution u = tan 1x>22 or equivalently x = 2 tan-1 u. The following relations are used in making this change of variables. A: dx = 2 1 + u2 du B: sin x = 2u 1 + u2 C: cos x = 1 - u2 1 + u2 Evaluate L p>2 0 du cos u + sin u .

8894. Rational functions of trigonometric functions An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution u = tan 1x>22 or equivalently x = 2 tan-1 u. The following relations are used in making this change of variables. A: dx = 2 1 + u2 du B: sin x = 2u 1 + u2 C: cos x = 1 - u2 1 + u2 Evaluate L p>3 0 sin u 1 - sin u du.

Three start-ups Three cars, A, B, and C, start from rest and accelerate along a line according to the following velocity functions: vA1t2 = 88t t + 1 , vB1t2 = 88t2 1t + 122, and vC1t2 = 88t2 t2 + 1 . a. Which car travels farthest on the interval 0 t 1? b. Which car travels farthest on the interval 0 t 5? c. Find the position functions for each car assuming that each car starts at the origin. d. Which car ultimately gains the lead and remains in front?

Skydiving A skydiver has a downward velocity given by v1t2 = VT a 1 - e-2gt>VT 1 + e-2gt>VT b, where t = 0 is the instant the skydiver starts falling, g _ 9.8 m>s2 is the acceleration due to gravity, and VT is the terminal velocity of the skydiver. a. Evaluate v102 and lim tS_ v1t2 and interpret these results. b. Graph the velocity function. c. Verify by integration that the position function is given by s1t2 = VT t + VT 2 g ln a 1 + e-2gt>VT 2 b, where s_1t2 = v1t2 and s102 = 0. d. Graph the position function. (See the Guided Project Terminal velocity for more details on free fall and terminal velocity.)

P * 22 7 One of the earliest approximations to p is 22 7 . Verify that 0 6 L 1 0 x411 - x24 1 + x2 dx = 22 7 - p. Why can you conclude that p 6 22 7 ?

Challenge Show that with the change of variables u = 1tan x, the integral 11tan x dx can be converted to an integral amenable to partial fractions. Evaluate 1 p>4 0 1tan x dx.