372 Higher Engineering Mathematics

7. (a)

∫

3cos2xdx (b)

∫

7sin3θdθ

⎡

⎢

⎢

⎣

(a)

3

2

sin2x+c

(b)−

7

3

cos3θ+c

⎤

⎥

⎥

⎦

8. (a)

∫

3

4

sec^23 xdx (b)

∫

2cosec^24 θdθ

[

(a)

1

4

tan3x+c(b)−

1

2

cot4θ+c

]

9. (a) 5

∫

cot 2tcosec 2tdt

(b)

∫

4

3

sec4ttan4tdt

⎡

⎢

⎢

⎣

(a)−

5

2

cosec2t+c

(b)

1

3

sec4t+c

⎤

⎥

⎥

⎦

10. (a)

∫

3

4

e^2 xdx (b)

2

3

∫

dx

e^5 x

[

(a)

3

8

e^2 x+c (b)

− 2

15e^5 x

+c

]

11. (a)

∫

2

3 x

dx (b)

∫(

u^2 − 1

u

)

du

[

(a)

2

3

lnx+c (b)

u^2

2

−lnu+c

]

12. (a)

∫

( 2 + 3 x)^2

√

x

dx (b)

∫(

1

t

+ 2 t

) 2

dt

⎡

⎢

⎢

⎣

(a) 8

√

x+ 8

√

x^3 +

18

5

√

x^5 +c

(b)−

1

t

+ 4 t+

4 t^3

3

+c

⎤

⎥

⎥

⎦

37.4 Definite integrals

Integrals containing an arbitrary constantc in their

results are calledindefinite integralssince theirprecise

value cannot be determined without furtherinformation.

Definite integralsare those in which limits are applied.

If an expression is written as [x]ba,‘b’ is called theupper

limitand ‘a’thelower limit. The operation of applying

the limits is defined as [x]ba=(b)−(a).

The increase in the value of the integralx^2 asxincreases

from 1 to 3 is written as

∫ 3

1 x

(^2) dx.
Applying the limits gives:
∫ 3
1
x^2 dx=
[
x^3
3
+c
] 3
1

(
33
3
+c
)
−
(
13
3
+c
)
=( 9 +c)−
(
1
3
+c
)
= 8
2
3
Notethatthe‘c’ term always cancels out when limitsare
applied and it need not be shown with definite integrals.
Problem 12. Evaluate
(a)
∫ 2
13 xdx (b)
∫ 3
− 2 (^4 −x
(^2) )dx.
(a)
∫ 2
1
3 xdx=
[
3 x^2
2
] 2
1

{
3
2
( 2 )^2
}
−
{
3
2
( 1 )^2
}
= 6 − 1
1
2
= 4
1
2
(b)
∫ 3
− 2
( 4 −x^2 )dx=
[
4 x−
x^3
3
] 3
− 2

{
4 ( 3 )−
( 3 )^3
3
}
−
{
4 (− 2 )−
(− 2 )^3
3
}
={ 12 − 9 }−
{
− 8 −
− 8
3
}
={ 3 }−
{
− 5
1
3
}
= 8
1
3
Problem 13. Evaluate
∫ 4
1
(
θ+ 2
√
θ
)
dθ,taking
positive square roots only.
∫ 4
1
(
θ+ 2
√
θ
)
dθ=
∫ 4
1
(
θ
θ
1
2

• 
  

  2
  θ
  1
  2
  )
  dθ

  
  

  ∫ 4
  1
  (
  θ
  1
  (^2) + 2 θ
  − 1
  2
  )
  dθ

  
  

  ⎡
  ⎢
  ⎣
  θ
  ( 1
  2
  )

  
  


• 1
  1
  2


• 1


• 
  

  2 θ
  (− 1
  2
  )

  
  


• 1
  −
  1
  2


• 1
  ⎤
  ⎥
  ⎦
  4
  1