WMA11 Jan 2021

9. WMA11/01 Edexcel IAL P1 January 2021 IAL Q9 (Integration)

• Find

∫3x+224xdx            x>0

giving your answer in simplest form.​​ 

(5)

• A curve C has equation y = f(x).​​ 

Given​​ 

• fʹ(x) = x2​​ + ax + b where a and b​​ are constants​​ 

• the y intercept of C is –8​​ 

• the point P(3,–2) lies on C​​ 

• the gradient of C at P is 2​​ 

find, in simplest form, f(x).​​ 

(6)

SOLUTION

i-​​ 

Let us first simplify the expression that needs to be integrated,​​ 

3x+224x12=9x2+12x+44x12

=9x24x12+12x4x12+44x12

=94x32+3x12+x-12

Now, let us integrate it.

∫94x32+3x12+x-12dx

=25x 94x52+23x 3x32+2x12+c

910x52+2x32+2x12+c 

Hence,​​ 

∫3x+224x=910x52+2x32+2x12+c 

ii-​​ 

We know that the integrationis anti-derivative. So,​​ 

fx=∫f'xdx

Substituting the expression of f’(x).

f(x)=∫x2+ax+bdx

fx=x33+ax22+bx +c

It is also given that the y-intercept is -8, which means that the constant term in the equation of a curve is -8.

fx=x33+ax22+bx-8

It is also given that the gradient of C at P (3, 2) is 2, so we will us the expression​​ f'x=x2+ax+b, and equate to 2 to get the value of​​ a.​​ 

f3= -2

33 +a322+b3-8 =-2

9+92a+3b -8=-2

92a+3b =-3

9a+6b =-6    (1)

Since curve passes through pont P, it must satisfy the equation.​​ 

3x2+a3+b =2

9+3a+b=2

3a+b=-7  -2

Multiplying equation (2) by 3, we get​​ 

9a+3b= -21     (3)

Subtracting equation 1 by 2.

9a+3b= -21     9a+6b =-6-            -              +

-3b=-21+6

b=-15-3

b=5

Where on substituting​​ the value of b in equation (1), we get the value of​​ a.​​ 

9a+6b =-6

9a+6(5) =-6

9a=-6-30

a=-369

a=-4

Now, going back to the expression of f(x), and substituting the values of​​ a​​ &​​ b.

fx=x33+ax22+bx-8

fx=x33-4x22+5x-8

fx=x33-2x2+5x-8