Download Integration Tables

INTEGRATION RULES
TRIGONOMETRIC INTEGRALS
Z
Z
Z
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(Af (x) + Bg (x)) dx = A f (x) dx + B g (x) dx
Z
f 0 (g (x)) g 0 (x) dx = f (g (x)) + C
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Z
U (x) dV (x) = U (x) V (x) −
a
Z
V (x) dU (x)
f 0 (x) dx = f (b) − f (a)
b
d
dx
Z
x
f (t) dt = f (x)
a
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Z
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Z
Z
Z
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Z
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xr dx =
Z
Z
Z
Z
ELEMENTARY INTEGRALS
Z
Z
1
xr+1 + C if r 6= −1
r+1
dx
= ln |x| + C
x
ex dx = ex + C
sin x dx = − cos x + C
cos x dx = sin x + C
sec2 x dx = tan x + C
csc2 x dx = − cot x + C
sec x tan x dx = sec x + C
csc x cot x dx = − csc x + C
tan x dx = ln |sec x| + C
cot x dx = ln |sin x| + C
sec x dx = ln |sec x + tan x| + C
csc x dx = ln |csc x − cot x| + C
dx
x
√
= sin−1 + C (a > 0, |x| < a)
a
a2 − x2
Z
1
x
dx
= tan−1 + C (a > 0)
a2 + x2
a
a
¯
¯
Z
¯x + a¯
dx
1
¯
¯ + C (a > 0)
ln
=
a2 − x2
2a ¯ x − a ¯
Z
¯x¯
1
dx
¯ ¯
√
= sec−1 ¯ ¯ + C (a > 0, |x| > a)
2
2
a
a
x x −a
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sin2 x dx =
1
x
− sin 2x + C
2
4
cos2 x dx =
x
1
+ sin 2x + C
2
4
tan2 x dx = tan x − x + C
cot2 x dx = − cot x − x + C
sec3 x dx =
1
1
sec x tan x + ln |sec x + tan x| + C
2
2
1
1
csc3 x dx = − csc x cot x + ln |csc x − cot x| + C
2
2
sin ax sin bx dx =
cos ax cos bx dx =
sin (a + b) x
sin (a − b) x
−
+ C if a2 6= b2
2 (a − b)
2 (a + b)
sin (a + b) x
sin (a − b) x
+
+ C if a2 6= b2
2 (a − b)
2 (a + b)
cos (a − b) x
cos (a + b) x
−
+ C if a2 6= b2
2 (a − b)
2 (a + b)
Z
1
n−1
sinn−2 x dx
sinn x dx = − sinn−1 x cos x +
n
n
Z
1
n−1
cosn−2 x dx
cosn x dx = cosn−1 x sin x +
n
n
Z
1
tann−1 x − tann−2 x dx if n 6= 1
tann x dx =
n−1
Z
−1
cotn x dx =
cotn−1 x − cotn−2 x dx if n 6= 1
n−1
Z
1
n−2
secn−2 x tan x +
secn−2 x dx if n 6= 1
secn x dx =
n−1
n−1
Z
−1
n−2
cscn−2 x cot x +
cscn−2 x dx if n 6= 1
cscn x dx =
n−1
n−1
Z
sinn−1 x cosm+1 x
n−1
sinn−2 x cosm x dx if n 6= −m
sinn x cosm x dx = −
+
n+m
n+m
Z
m−1
sinn+1 x cosm−1 x
sinn x cosm x dx =
+
sinn x cosm−2 x dx if m 6= −n
n+m
n+m
sin ax cos bx dx = −
x sin x dx = sin x − x cos x + C
x cos x dx = cos x + x sin x + C
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xn sin x dx = −xn cos x + n xn−1 cos x dx
xn cos x dx = xn sin x − n
Z
xn−1 sin x dx
TRIGONOMETRIC IDENTITIES
sin2 x + cos2 x = 1
sin (−x) = − sin x
cos(−x) = cos x
sec2 x = 1 + tan2 x
sin (π − x) = sin x
cos(π − x) = − cos x
csc2 x = 1 + cot2 x
sin
sin (x ± y) = sin x cos y ± cos x sin y
cos (x ± y) = cos x cos y ∓ sin x sin y
sin 2x = 2 sin x cos x
sin2 x =
³π
2
´
− x = cos x
1 − cos 2x
2
QUADRATIC FORMULA
x=
−B ±
³π
2
´
− x = sin x
tan (x ± y) =
cos2 x =
cos 2x = 2 cos2 x − 1 = 1 − 2 sin2 x
Ax2 + Bx + C = 0
cos
√
B 2 − 4AC
2A
2
tan x ± tan y
1 ∓ tan x tan y
1 + cos 2
2
INTEGRALS INVOLVING
(If
Z
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Z
Z
Z
Z
Z
Z
√
x2 ± a2
(a > 0)
INTEGRALS INVOLVING
√
x2 − a2 , assume |x| > a > 0.)
¯
√
√
x√ 2
a2 ¯¯
¯
ln ¯x + x2 ± a2 ¯ + C
x2 ± a2 dx =
x ± a2 ±
2
2
¯
¯
√
dx
¯
¯
√
= ln ¯x + x2 ± a2 ¯ + C
2
2
x ±a
¯
¯
√
¯ a + √x2 + a2 ¯
√
x2 + a2
¯
¯
2
2
dx = x + a − a ln ¯
¯+C
¯
¯
x
x
√
√
2
2
2
2
√
x −a
x −a
dx = x2 − a2 − a tan−1
+C
x
a
¯
√
√
¢√
x¡ 2
a4 ¯¯
¯
2x ± a2
ln ¯x + x2 ± a2 ¯ + C
x2 ± a2 −
x2 x2 ± a2 dx =
8
8
¯
√
x2
x√ 2
a2 ¯¯
¯
√
dx =
x ± a2 ∓
ln ¯x + x2 ± a2 ¯ + C
2
2
2
2
x ±a
√
√
¯
¯
√
x2 ± a2
x2 ± a2
¯
¯
+ ln ¯x + x2 ± a2 ¯ + C
dx
=
−
2
x
x
√
x2 ± a2
dx
+C
√
=∓
2
2
2
a2 x
x x ±a
dx
(x2 ± a2 )3/2
¡
x2 ± a2
¢3/2
=
±x
√
+C
a2 x2 ± a2
dx =
¯
√
¢√
x¡ 2
3a4 ¯¯
¯
x2 ± a2 +
2x ± 5a2
ln ¯x + x2 ± a2 ¯ + C
8
8
√
a2 − x2
(a > 0, |x| < a)
Z
√
x√ 2
a2
x
sin−1 + C
a2 − x2 dx =
a − x2 +
2
2¯
a
¯
Z √ 2
¯ a + √a2 − x2 ¯
√
a − x2
¯
¯
dx = a2 − x2 − a ln ¯
¯+C
¯
¯
x
x
Z
x
x√ 2
a2
x2
√
dx = −
a − x2 +
sin−1 + C
2
2
a
a2 − x2
Z
√
¢√
x
x¡ 2
a4
a2 − x2 +
x2 a2 − x2 dx =
2x − a2
sin−1 + C
8
8
a
√
Z
a2 − x2
dx
+C
√
=−
a2 x
x2 a2 − x2
√
√
Z
a2 − x2
a2 − x2
x
dx = −
− sin−1 + C
2
x
x
a
¯
¯
√
Z
1 ¯¯ a + a2 − x2 ¯¯
dx
√
= − ln ¯
¯+C
¯
a ¯
x
x a2 − x2
Z
x
dx
=
√
+C
a2 a2 − x2
(a2 − x2 )3/2
Z
¢3/2
¢√
¡ 2
x
x¡ 2
3a4
a − x2
dx =
a2 − x2 +
5a − 2x2
sin−1 + C
8
8
a
INTEGRALS OF INVERSE TRIGONOMETRIC FUNCTIONS
EXPONENTIAL AND LOGARITHMIC INTEGRALS
Z
Z
xex dx = (x − 1) ex + C
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ln x dx = x ln x − x + C
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sin−1 x dx = x sin−1 x +
√
1 − x2 + C
¢
1 ¡
ln 1 + x2 + C
2
Z
¯
¯
√
¯
¯
sec−1 x dx = x sec−1 x − ln ¯x + x2 − 1¯ + C
Z
tan−1 x dx = x tan−1 x −
(x > 1)
Z
Z
xn ex dx = xn ex − n xn−1 ex dx
xn+1
xn+1
ln x −
+ C,
(n 6= −1)
n+1
(n + 1)2
Z
Z
xn+1
m
xn (ln x)m−1 dx
xn (ln x)m dx =
(ln x)m −
n+1
n+1
Z
Z
xn ln x dx =
(n 6= −1)
eax sin bxdx =
eax cos bxdx =
eax
a2 + b2
eax
a2 + b2
(a sin bx − b cos bx) + C
(a cos bx + b sin bx) + C
C :\a ll\crss\IntTa b le-S ta stn a .te x
3