Examples on Integrals Involving Inverse Trigonometric Functions

Examples I

1.

dx

x

x.

1 8

3

cx

dxx

x

dxx

x

441

8

3

41

8

3

arcsin

.1

4

.1

2.

184

3

xx

dxx

cxarc

xx

dxx

xx

dxx

441

84

3

41

84

3

sec

1

4

1

2*

18xx

dx

cxarc

xx

dxx

xx

dx

441

84

3

41

8

sec

1

4

1

3.

dx

x

x.

1 8

3

cx

dxx

x

dxx

x

arctan

.1

4

.1

41

8

3

41

8

3

Examples II

1.

dx

x

x.

49 8

3

cx

dxx

x

dxx

x

dxx

x

)arcsin(

.)(1

.)(1

.49

432

81

2432

338

31

83

2432

3

31

8

3

49 8xx

dx

cxarc

xx

dxx

xx

dxx

xx

dxxx

dx

)sec(

1)(

4

1)(

1)(

49

423

81

24234

23

323

21

41

24234

3

21

2423

21

8

3.

dx

x

x.

34 8

3

cx

dxx

x

dxx

x

dxx

x

)arctan(

.)(1

4

.)(1

.34

423

381

2423

323

41

41

32

2423

3

41

8

3

Examples III

1.

dx

x

x.

2cot9

2csc2

2

cx

dxx

x

dxx

x

dxx

x

)2cotarctan(

.)2cot(1

2)2csc(

.)2cot(1

2csc

.2cot9

2csc

31

61

231

231

91

23

231

2

91

2

2

2.

dx

x

x.

5cos16

5sin2

cx

dxx

x

dxx

x

dxx

x

)5cosarcsin(

.)5cos(1

5)5sin(

.)5cos(1

5sin

.5cos16

5sin

41

51

241

41

41

54

241

41

2

3.

dxx

xx.

7sec225

7tan7sec2

cx

dxx

xx

dxx

xx

dxx

xx

)7secarctan(

)7sec(1

7)7tan7(sec

.)7sec(1

7tan7sec

.7sec225

7tan7sec

52

2351

252

52

251

275

25225

1

2

Examples IV

1.

dx

e

ex

x

.54 6

3

ce

dxe

e

dxe

e

dxe

e

x

x

x

x

x

x

x

)arctan(

.)(1

3

.)(1

.54

325

561

2325

325

41

532

2325

3

41

6

3

2.

dx

e

ex

x

.59 6

3

ce

dxe

e

dxe

e

dxe

e

x

x

x

x

x

x

x

)arcsin(

.)(1

3

.)(1

.59

335

531

2335

335

31

51

2335

3

31

6

3

3.

95 6xe

dx

cearc

ee

dxe

e

dxe

dx

x

xx

x

x

x

)sec(

1)(

3

1)(

95

335

91

23353

35

335

31

31

2335

31

6

Caution!!

Do not Confuse the arcsin formula with the power formula!

Example 1

Notice the difference between the previous integral

And the integral

dx

x

x.

1 8

3

dx

x

x.

1 8

7

cx

dxxx

dxxx

dxx

x

21

8

81

7881

78

8

7

21

21

21

)1(

)8()1(

)1(

.1

Do not Confuse the arcsin formula with the power formula!

Example 2

Notice the difference between the previous integral

And the integral

dxx

x.

5cos16

5sin

cxdxx

x

)5cosarcsin(.5cos16

5sin41

51

2

cx

dxxx

dxxx

dxx

x

215

1

51

21

21

21

)5cos16(

55sin)5cos16(

5sin)5cos16(

.5cos16

5sin

Do not Confuse the arcsin formula with the power formula!

Example 3

Notice the difference between the previous integral

and the integral

cedxe

e x

x

x

)arcsin(.59

335

531

6

3

dx

e

ex

x

3

3

59

ce

dxee

dxee

dxe

e

x

xx

xx

x

x

21

3

151

33151

33

3

3

21

21

21

)59(

)3()5()59(

)59(

59

Do not Confuse the arctan formula with the logarithmic formula!

Example 1

Notice the difference between the previous integral

And the integral

cxdxx

x

441

8

3

arctan.1

dx

x

x.

1 8

7

cx

cx

dxx

x

dxx

x

)1ln(

1ln

.1

8

.1

881

881

8

7

81

8

7

Do not Confuse the arctan formula with the logarithmic formula!

Example 2Notice the difference between the previous

integral

And the integral

Or the integral

cxdxx

x

)2cotarctan(.2cot9

2csc31

61

2

2

dx

x

x.

2cot9

2csc2

dx

x

xx.

2cot9

2csc2cot2

2

cx

dxx

x

dxx

x

2cot9ln

.2cot9

2csc2

.2cot9

2csc

21

2

21

2

cx

dxx

xx

dxx

xx

2cot9ln

.2cot9

2csc2cot4

.2cot9

2csc2cot

241

2

2

41

2

2

Do not Confuse the arctan formula with the logarithmic formula!

Example 3Notice the difference between the previous integral

and the integral

dx

e

ex

x

.54 3

3

cedxe

e xx

x

)arctan(.54

325

561

6

3

ce

ce

dxe

e

dxe

e

x

x

x

x

x

x

)54ln(

54ln

.54

15

.54

3151

3151

3

3

151

3

3