Denition approach of learning new topics

30 5 INTEGRATION
Part IV
Lecture 4
5 Integration
5.1 Dierentials
Let us rst understand what do we mean by dierentials
Dierential of x is ∆x and is denoted as dx
lim
∆x→0
So a dierential dx is a small dynamic change in x (Remember dierential
of x is d(x) = dx)
Example. See the following worked examples of dierentials
1. Dierential of y is dy
2. Dierential of f(x) is d(f(x)) = f ′ (x)dx
3. Dierential of sin x is d(sin x) = cos x dx
4. Dierential of sin x 2 is d(sin x 2 ) = cos x 2 2x dx
5. Dierential of f(g(x)) is d(f(g(x))) = f ′ (g(x)) g ′ (x)dx
This can be seen in a dierent way, dierential of f(x) is d(f(x)) this can be
produced by dierenting f(x) with respect to the independent variable here i.e.
x
d df(x)
f(x) =
dx dx
= f ′ (x) ⇒ d(f(x)) = f ′ (x) dx
5.2 Indenite Integration
Denition. It is dened as reverse operation of Dierentiation.
Anti-derivative.
Also called
Notation: ´
f(x)dx : read as integral of f(x) dx which means we are trying to nd
that function whose derivative with respect to x is f(x)
for example: ´ sin x dx = − cos x : means derivative of − cos x is sin x
1.
2.
3.
4.
d
dx (xn+1 ) = (n + 1) · x n ↔ ´ x n dx = xn+1
n + 1
d
dx (ax ) = a x · log e a ↔ ´ a x dx =
ax
log e a
d
dx (log e x) = 1 x ↔´ 1
x dx = log e x
d
dx (ex ) = e x ↔ ´ e x dx = e x
5. Trigonometric functions
where n ≠ −1