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Кузбасский государственный технический университет

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[НЕСОРТИРОВАННОЕ]

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6.2. INTEGRATION BY SUBSTITUTION

273

csc t(csc t − cot t)dt

sin x

20.

21.

cos2 x

sin2 x

sin2 t

cos2 t

22.

cos x

23.

sin3 t − 3

24.

cos3 t + 2

25.

tan2 t dt

26.

cot2 t dt

27.

Z (2 sec2 t + 1)dt

28.

29.

sinh t dt

30.

cosh t dt

31.Determine f(x) if f0(x) = cos x and f(0) = 2.

32.Determine f(x) if f00(x) = sin x and f(0) = 1, f0(0) = 2.

33.Determine f(x) if f00(x) = sinh x and f(0) = 2, f0(0) = −3.

34.Prove each of the integration formulas 1–77.

6.2Integration by Substitution

Theorem 6.2.1 Let f(x), g(x), f(g(x)) and g0(x) be continuous on an interval [a, b]. Suppose that F 0(u) = f(u) where u = g(x). Then

(i)	Z	f(g(x))g0(x)dx = Z	f(u)du = F (g(x)) + C
			u=g(b)
(ii)	Zab f(g(x))g0(x)dx = Zu=g(u)			f(u)du = F (g(b)) − F (g(a)).

Proof. See the proof of Theorem 5.3.1.

Exercises 6.2 In problems 1–39, evaluate the integral by making the given substitution.

274

CHAPTER 6.

TECHNIQUES OF INTEGRATION

3x(x2 + 1)10dx, u = x2 + 1

x sin(1 + x2)dx, u = 1 + x2

√

dt, x = √

√t

(1 + x3)3/2 dx, u = 1 + x3

cos( t)

2earcsin x

3earccos x

√

dx, u = arcsin x

√

1 − x2

x 4x2 dx, u = 4x2

10sin x cos x dx, u = sin x

4arctan x

(1 + ln x)10

dx, u = 4arctan x

10.

dx, u = 1 + ln x

1 + x2

5arcsec x

Z (tan 2x)3 sec2 2x dx, u = tan 2x

11.

x√

dx, u = arcsec x

12.

x2 − 1

13.

(cot 3x)5 csc2 3x dx, u = cot 3x

14.

sin21 x cos x dx, u = sin x

15.

cos5 x sin x dx, u = cos x

16.

(1 + sin x)10 cos x dx, u = 1 + sin x

17.

sin3 x dx, u = cos x

18.

cos3 x dx, u = sin x

19.

tan3 x dx, u = tan x

20.

cot3 x dx, u = cot x

21.

sec4 x dx, u = tan x

22.

csc4 x dx, u = cot x

23.

sin3 x cos3 x dx, u = sin x

24.

sin3 x cos3 x dx, u = cos x

25.

tan4 x dx, u = tan x

26.

sin(ln x)

dx, u = ln x

6.2.

INTEGRATION BY SUBSTITUTION

275

27.

1 + x2

dx, u = ln(1 + x)2

28.

tan3 x sec4 x dx, u = sec x

x cos(ln(1 + x2))

29.

cot3 x csc4 x dx, u = csc x

30.

√

, x = 2 sin t

4 x2

√9 − x2

−

31.

, x = 3 cos t

32.

√4 + x2

, x = 2 sinh t

33.

√x2 − 9

, x = 3 cosh t

34.

4 + x2 , x = 2 tan t

35.

4 − x2 , x = 2 tanh t

36.

x√x2 − 4

, x = 2 sec t

37.

4esin(3x) cos(3x)dx, u = sin 3x

38.

x 3(x2+4)dx, u = 3x2+4

3 etan 2x sec2 x dx, u = tan 2x

x√

dx, u = x + 2

39.

40.

x + 2

Evaluate the following deﬁnite integrals.

41.

(x + 1)30dx

42.

2 x(4 − x2)1/2dx

43.

π/4 tan3 x sec2 x dx

44.

1 x3(x2 + 1)3dx

45.

(x + 1)(x − 2)10dx

46.

8 x2(1 + x)1/2dx

47.

π/6 sin(3x)dx

48.

π/4 cos(2x)dx

49.

π/4 sin3 2x cos 2x dx

50.

π/6 cos4 3x sin 3x dx

earctan x

1/2

earcsin x

51.

52.

√

1 + x2

1 − x2

276					CHAPTER 6. TECHNIQUES OF INTEGRATION
	3	earcsec x				1		dx
53.	Z2	x√		dx	54.	Z0	√
			x2 − 1					1 + x2

6.3Integration by Parts

Theorem 6.3.1 Let f(x), g(x), f0(x) and g0(x) be continuous on an interval [a, b]. Then


(i)	Z	f(x)g0	(x)dx = f(x)g(x) − Z		g(x)f0(x)dx
(ii)	Zab f(x)g0		(x)dx = (f(b)g(b) − f(a)g(a)) − Zab g(x)f0(x)dx
(iii)	Z	udv = uv − Z		vdu

where u = f(x) and dv = g0(x)dx are the parts of the integrand. Proof. See the proof of Theorem 5.4.1.

Exercises 6.3 Evaluate each of the following integrals.

1.	Z	x sin x dx	2.	Z	x cos x dx
3.	Z	x ln x dx	4.	Z	x ex dx
5.	Z	x 4x dx	6.	Z	x2 ln x dx
7.	Z	x2 sin x dx	8.	Z	x2 cos x dx
9.	Z	x2ex dx	10.	Z	x2 10x dx

6.3.	INTEGRATION BY PARTS				277
11.	Z	ex sin x dx (Let u = ex twice and solve.)
12.	Z	ex cos x dx (Let u = ex twice and solve.)
13.	Z	e2x sin 3x dx (Let u = e2x twice and solve.)
14.	Z	x sin(3x)dx	15.	Z	x2 cos(2x)dx
16.	Z	x2e4xdx	17.	Z	x3 ln(2x)dx
18.	Z	x sec2 x dx	19.	Z	x csc2 x dx
20.	Z	x sinh(4x)dx	21.	Z	x2 cosh x dx
22.	Z	x cos(5x)dx	23.	Z	sin(ln x)dx
24.	Z	cos(ln x)dx	25.	Z	x arcsin x dx
26.	Z	x arccos x dx	27.	Z	x arctan x dx
28.	Z	x arcsec x dx	29.	Z	arcsin x dx
30.	Z	arccos x dx	31.	Z	arctan x dx
	Z

32.arcsec x dx

Verify the following integration formulas:

(sinn−2

278	CHAPTER 6. TECHNIQUES OF INTEGRATION

33.

sinn(ax)dx =

−

sinn−1

(ax) cos(ax)

n − 1

34.

cosn(ax)dx =

cosn−1(ax) sin(ax) +

n − 1

35.

xnexdx = xnex − n Z

xn−1exdx

ax)dx

(cosn−2 ax)dx

36.

xn sin x dx = −xn cos x + n

xn−1 cos x dx

37.

xn cos x dx = xn sin x − n Z

xn−1 sin x dx

38.

eax sin(bx)dx =

eax[a sin(bx) − b cos(bx)] + C

a2 + b2

39.

eax cos(bx) dx =

eax[a cos(bx) + b sin(bx)] + C

a2 + b2

40.

xn ln x dx =

xn+1 ln x −

xn+1 + C, n 6= −1, x > 0

n + 1

(n +

1)2

41.

secn x dx =

secn−2 x tan x +

n − 2

secn−2 x dx, n = 1, n > 0

n − 1

42.

cscn x dx =

−1

cscn−2 x cot x +

n − 2

cscn−2 x dx, n = 1, n > 0

n − 1

Use the formulas 33–42 to evaluate the following integrals:

43.

sin4 x dx

44.

cos5 x dx

45.

x3exdx

46.

x4 sin x dx

47.

x3 cos x dx

48.

e2x sin 3x dx

49.

e3x cos 2x dx

50.

x5 ln x dx

6.3.	INTEGRATION BY PARTS							279
51.	Z	sec3 x dx			52.		Z	csc3 x dx
Prove each of the following formulas:
53.	Z	tann x dx =	1		tann−1 x − Z	tann−2 x dx, n 6= 1

				n − 1
54.	Z	cotn x dx =	1		cotn−1 x − Z	cotn−2 x dx, n 6= 1

			n − 1
55.	Z	sin2n+1 x dx = − Z			(1 − u2)ndu, u = cos x

56.cos2n+1 x dx = − (1 − u2)ndu, u = sin x

57.	Z	sin2n+1 x cosm x dx = − Z	(1 − u2)numdu, u = cos x
58.	Z	cos2n+1 x sinm x dx = Z (1 − u2)numdu, u = sin x
	Z	Z

59.sin2n x cos2m x dx = (sin x)2n(1 − sin2 x)mdx

60.	Z	tann x sec2m x dx =	Z	un(1 + u2)m−1du, u = tan x
61.	Z	cotn x csc2m x dx = − Z			un(1 + u2)m−1du, u = cot x
62.	Z	tan2n+1 x secm x dx = Z (u2 − 1)num−1du, u = sec x
63.	Z	cot2n+1 x cscm x dx = − Z (u2 − 1)num−1du, u = csc x
64.	Z	sin mx cos nx dx = −2			m + n	+	m	−n	+ C; m2	6= n2
			1		cos(m + n)x		cos(m	n)x

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