Page 12 - Formula-All

B. Tech. Semester-I (All Branch) Important Formulas

(D) Standard Results for fixed arrangement:

Z 0
f (x)
(1) dx = log |f(x)| + c
f(x)

Z [f(x)] n+1
0
n
(2) [f(x)] f (x) dx = + c
n + 1

Z
x
0
x
(3) e [f(x) + f (x)] dx = e f(x) + c

Z e ax
(4) e ax sin bx dx = [a sin bx − b cos bx] + c
2
a + b 2

Z ax
e
(5) e ax cos bx dx = [a cos bx + b sin bx] + c
2
a + b 2

(E) Integration and Differentiation together(Leibnitz’s Rule):
Z
00
000
0
u · v dx = u · v − u · v + u · v − u · v + · · · · · · · · ·
2
3
3
1
(F) Definite Integration rules
b
Z
b
(1) f(x) dx = [F(x)] = F(b) − F(a)
a
a
b a
Z Z
(2) f(x) dx = − f(x) dx

a b

b c c
Z Z Z
(3) f(x) dx + f(x) dx = f(x) dx

a b a


a 0, if f(x) is odd
Z

(4) f(x) dx = a
R
2 f(x) dx, if f(x) is even

−a
0
a a 2a a a
Z Z Z Z Z
(5) f(x) dx = f(a − x) dx (6) f(x) dx = f(x) dx + f(2a − x) dx

0 0 0 0 0

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