Understanding Engineering Mathematics

As usual, the places where you might have trouble occur when negative

signs and fractions are involved. In such cases, just take it steady and

check each step.

(i)

∫

3 dx= 3

∫

dx= 3 x+C

(ii)

∫

2

x^2

dx= 2

∫

x−^2 dx= 2

x−^2 +^1

− 2 + 1

+C= 2

x−^1

− 1

+C=−

2

x

+C

(iii)

∫

√

xdx=

∫

x

1

(^2) dx=
x
1
2 +^1
1
2 +^1
+C=
x^3 /^2
3 / 2
+C=
2
3
x^3 /^2 +C
(iv)
∫
x
1
(^3) dx=
x
1
3 +^1
1
3 +^1
+C=
x^4 /^3
4 / 3
+C=
3
4
x^4 /^3 +C
B. These are all standard integrals, known because we know what to
differentiate to get them.
(i)
d
dx
(cosx)=−sinx
so
∫
sinxdx=−cosx+C
(ii)
d
dx
(ex)=ex
so
∫
exdx=ex+C
(iii)
d
dx
(ln|x|)=
1
x
so
∫
1
x
dx=ln|x|+C
NB Note that
∫
1
x
dx =
x^0
0
! Also note the reminder about the modulus
in ln|x|.
These and others we can put together in the table of standard integrals
on page 256.
9.2.3 Addition of integrals
➤
251 281➤
If you differentiate a sum then the result is the sum of the derivatives:
d
dx
(f+g)=
df
dx

• dg
  dx
  Similarly, the integral of a sum is the sum of the integrals:
  ∫
  (f+g)dx=
  ∫
  fdx+
  ∫
  gdx
  (not forgetting the arbitrary constant of course).