Higher Engineering Mathematics

398 INTEGRAL CALCULUS

Table 40.1 Integrals using trigonometric and hyperbolic substitutions

f(x)

∫

f(x)dx Method See problem

1. cos^2 x

1

2

(

x+

sin 2x

2

)

+c Use cos 2x=2 cos^2 x− 1 1

2. sin^2 x

1

2

(

x−

sin 2x

2

)

+c Use cos 2x= 1 −2 sin^2 x 2

3. tan^2 x tanx−x+c Use 1+tan^2 x=sec^2 x 3


4. cot^2 x −cotx−x+c Use cot^2 x+ 1 =cosec^2 x 4


5. cosmxsinnx (a) If eithermornis odd (but not both), use
   cos^2 x+sin^2 x= 1 5, 6
   (b) If bothmandnare even, use either
   cos 2x=2 cos^2 x−1 or cos 2x= 1 −2 sin^2 x 7, 8


6. sinAcosB Use^12 [ sin(A+B)+sin(A−B)] 9


7. cosAsinB Use^12 [ sin(A+B)−sin(A−B)] 10


8. cosAcosB Use^12 [ cos(A+B)+cos(A−B)] 11


9. sinAsinB Use−^12 [ cos(A+B)−cos(A−B)] 12

10.

1

√

(a^2 −x^2 )

sin−^1

x

a

+c Usex=asinθsubstitution 13, 14

11.

√

(a^2 −x^2 )

a^2

2

sin−^1

x

a

+

x

2

√

(a^2 −x^2 )+c Usex=asinθsubstitution 15, 16

12.

1

a^2 +x^2

1

a

tan−^1

x

a

+c Usex=atanθsubstitution 17–19

13.

1

√

(x^2 +a^2 )

sinh−^1

x

a

+c Usex=asinhθsubstitution 20–22

or ln

{

x+

√

(x^2 +a^2 )

a

}

+c

14.

√

(x^2 +a^2 )

a^2

2

sinh−^1

x

a

+

x

2

√

(x^2 +a^2 )+c Usex=asinhθsubstitution 23

15.

1

√

(x^2 −a^2 )

cosh−^1

x

a

+c Usex=acoshθsubstitution 24, 25

or ln

{

x+

√

(x^2 −a^2 )

a

}

+c

16.

√

(x^2 −a^2 )

x

2

√

(x^2 −a^2 )−

a^2

2

cosh−^1

x

a

+c Usex=acoshθsubstitution 26, 27