Enzyme, Pharmaceutical Aids, Miscellaneous Last Part of Chapter no 5th.pdf
Integral calculus
1. INTEGRAL CALCULUS
I B.Sc Mathematics
18UMTC21
Mrs. P. Kalai Selvi ,M.Sc., M.Phil.,
Ms. M. Indira Devi, M.Sc., M.Phil.,
2. Anti-derivative
If f(x) is a continuous function and F(x) is the function
whose derivative is f(x), i.e.: 𝑭′
(x) = f(x) .
then:
𝒇 𝒙 𝒅𝒙 = F(x) + c;
where c is any arbitrary constant.
5. For example, to integrate 4x, we will write it as
follows:
4𝑥 𝒅𝒙 = 2x2
+ c , c ∈ ℝ.
Integral
sign
This term is called
the integrand
There must always be
a term of the form dx
Constant of
integration
6. 0
sinI xdx
00
sin cos
cos cos0
( 1) ( 1) 2
I xdx x
Another example with limit value:
( )f x( ) ( )F x f x dx ( )af x( )aF x( ) ( )u x v x( ) ( )u x dx v x dx
aax 1n
x n 1
1
n
x
n
ax
eax
e
a
1
x
ln xsin ax1
cosax
a
cosax1
sin ax
a
2
sin ax1 1
sin 2
2 4
x ax
a
7. TABLE OF INTEGRATION FORMULAS
1
1
1. ( 1) 2. ln | |
1
3. 4.
ln
n
n
x
x x x
x
x dx n dx x
n x
a
e dx e a dx
a
8. 2 2
5. sin cos 6. cos sin
7. sec tan 8. csc cot
9. sec tan sec 10. csc cot csc
11. sec ln sec tan 12. csc ln csc cot
x dx x x dx x
x dx x x dx x
x x dx x x x dx x
x dx x x x dx x x
9. 1 1
2 2 2 2
13. tan ln sec 14. cot ln sin
15. sinh cosh 16. cosh sinh
1
17. tan 18. sin
x dx x x dx x
x dx x x dx x
dx x dx x
x a a a aa x
11. Let dv be the most complicated part of the
original integrand that fits a basic integration
Rule (including dx). Then u will be the remaining
factors.
(OR)
Let u be a portion of the integrand whose
derivative is a function simpler than u. Then dv
will be the remaining factors (including dx).
12. x
xe dxFor example:
x x x
xe dx xe e dx
x x x
xe dx xe e C
u = x dv= exdx
du = dx v = ex
13. 2
sinx xdx
u = x2 dv = sin x dx
du = 2x dx v = -cos x
2 2
sin cos 2 cosx xdx x x xdx
u = 2x dv = cos x dx
du = 2dx v = sin x
2 2
sin cos 2 sin 2sinx xdx x x x x xdx
2 2
sin cos 2 sin 2cosx xdx x x x x x C
For example :