P1^
6
Reversing(^) differentiation
The rule for integrating xn
Recall the rule for differentiation:
y = xn ⇒ d
d
y
x
= nxn^ −^1.
Similarly y = xn^ +^1 ⇒ d
d
y
x
= (n + 1)xn
or y = (^) ()n^1 + 1 xn^ +^1 ⇒ d
d
y
x
= xn.
Reversing this, integrating xn gives (^) nx
n+
- 1
1
.
This rule holds for all real values of the power n except –1.Note
In words: to integrate a power of x, add 1 to the power and divide by the new power.
This works even when n is negative or a fraction.! Differentiating^ x^ gives^ 1,^ so^ integrating^1 gives^ x.^ This^ follows^ the^ pattern^ if^ you^
remember that 1 = x^0.EXAMPLE 6.1 Given that d
d
y
x(^) = 3 x 2 + 4 x + 3
(i) find the general solution of this differential equation
(ii) find the equation of the curve with this gradient function which passes
through (1, 10).
SOLUTION
(i) By integration, y = 33 42 3
xx^32
- xc+^
= x^3 + 2 x 2 + 3 x + c, where c is a constant.
(ii) Since the curve passes through (1, 10),
10 = 13 + 2(1)^2 + 3(1) + c
c = 4
⇒ y = x^3 + 2 x^2 + 3 x + 4.
- xc+^