666 Higher Engineering Mathematics
Integral Calculus
Standard integrals:
y
∫
ydx
axn a
xn+^1
n+ 1
+c
(except wheren=−1)
cosax
1
a
sinax+c
sinax −
1
a
cosax+c
sec^2 ax
1
a
tanax+c
cosec^2 ax −
1
a
cotax+c
cosecaxcotax −
1
a
cosecax+c
secaxtanax
1
a
secax+c
eax
1
a
eax+c
1
x
lnx+c
tanax
1
a
ln(secax)+c
cos^2 x
1
2
(
x+
sin2x
2
)
+c
sin^2 x
1
2
(
x−
sin2x
2
)
+c
tan^2 x tanx−x+c
cot^2 x −cotx−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
∫
ydx
1
(a^2 +x^2 )
1
a
tan−^1
x
a
+c
1
√
(x^2 +a^2 )
sinh−^1
x
a
+cor
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
+cor
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
acosθ+bsinθ+c
dθlet
sinθ=
2 t
( 1 +t^2 )
cosθ=
1 −t^2
1 +t^2
and
dθ=
2 dt
( 1 +t^2 )
Integration by parts:
Ifuandvare both functions ofxthen:
∫
u
dv
dx
dx=uv−
∫
v
du
dx
dx
Reduction formulae:
∫
xnexdx=In=xnex−nIn− 1
∫
xncosxdx=In=xnsinx+nxn−^1 cosx
−n(n− 1 )In− 2