666 Higher Engineering Mathematics
Integral Calculus
Standard integrals:
y
y dx
ax
n
a
x n+1
n + 1
+ c
(except where n = −1)
cos ax
1
a
sin ax + c
sin ax
−
1
a
cos ax + c
sec 2 ax
1
a
tan ax + c
cosec 2 ax
−
1
a
cot ax + c
cosec ax cot ax −
1
a
cosec ax + c
sec ax tan ax
1
a
sec ax + c
e ax
1
a
e ax + c
1
x
ln x + c
tan ax
1
a
ln(sec ax) + c
cos 2 x
1
2
x +
sin 2x
2
+ c
sin 2 x
1
2
x −
sin 2x
2
+ c
tan 2 x
tan x − x + c
cot 2 x
−cot x − x + c
1
(a 2 − x 2 )
sin −1 x
a
+ c
(a 2 − x 2 )
a 2
2
sin −1 x
a
+
x
2
(a 2 − x 2 ) + c
y
y dx
1
(a 2 + x 2 )
1
a
tan −1 x
a
+ c
1
(x 2 + a 2 )
sinh
−1 x
a
+ c or
ln
x +
(x 2 + a 2 )
a
+ c
(x 2 + a 2 )
a 2
2
sinh −1 x
a
+
x
2
(x 2 + a 2 ) + c
1
(x 2 − a 2 )
cosh −1 x
a
+ c or
ln
x +
(x 2 − a 2 )
a
+ c
(x 2 − a 2 )
x
2
(x 2 − a 2 ) −
a 2
1
cosh
−1 x
a
+ c
t = tan
θ
2
substitution
To determine
1
a cos θ + b sin θ + c
dθ let
sin θ =
2t
(1 + t 2 )
cos θ =
1 − t 2
1 + t 2 and
dθ =
2 dt
(1 + t 2 )
Integration by parts:
If u and v are both functions of x then:
u
dv
dx
dx = uv −
v
du
dx
dx
Reduction formulae:
x
n e
x dx = I n = x
n e
x
− n I n−1
x
n cos x dx = I n = x
n sin x + nx
n−1 cos x
−n(n − 1)I n−2