Understanding Engineering Mathematics

9.3.3 Addition of integrals

(i)

2

3

x^3 +

3

2

1

x^2

(ii) 2 sinx+cosx (iii) sinhx=^12 (ex−ex)

(iv) −cos(x+ 3 ) (v)

∑n

r= 0

arxr+^1

r+ 1

(vi)

1

2

(

cosx−

1

3

cos 3x

)

(vii) coshx

9.3.4 Simplifying the integrand

A. (i)

x^2

2

−x(x    = 2 ) (ii) sinx (iii) sinx+cosx,provided cosx−sinx  = 0

(iv) −

2 e

x

(x > 0 ) (v) x+

1

2

cos 2x

B.ln|secx+tanx|,ln|cosecx−cotx|

9.3.5 Linear substitution in integration

(i)

1

32

( 4 x+ 3 )^8 (ii)

3

8

( 2 x− 1 )^4 /^3 (iii) −^13 cos( 3 x− 1 ) (iv)^12 e^4 x+x

(v)

3

2

ln( 2 x+ 1 ) (vi)

1

10

( 2 x− 1 )^5 (vii)

2

9

( 3 x+ 2 )^3 /^2 (viii)^12 sin( 2 x+ 1 )

(ix)

4

3

e^3 x−^1 (x)

1

4

ln( 4 x− 3 )

9.3.6 Thedu=f′.x/dxsubstitution

A. (i) u=x^3 ,^13 ex
3
(ii) u=tanx+2,−cos(tanx+2)

(iii) u=sinx,

sin^4 x

4

(iv) u=x^2 + 2 x+3,

1

2

ln(x^2 + 2 x+3)

(v) u=cosx,ln|secx|

B. (i) −cos(f (x)) (ii)^12 e(f (x))

2

(iii)^13 sin(x^3 + 1 )

(iv) cosxln(cosx)−cosx (v) ln(x− 1 ) (vi) ef(x)

(vii) sin(f (x)+ 2 )

C. (i)

1

4

(x^3 + 2 )^4 (ii)

2

3

(x^2 + 1 )^3 /^2 (iii) ln(x^2 −x+ 1 )

(iv)

1

2

sin(x^2 + 2 x+ 1 ) (v) −ecosx (vi) ln(tanx)

(vii)

1

16

(2sin2x+ 1 )^4 (viii) 6

√

x^2 + 4 x+ 4