Category Archives: Integration [O Level]

Solving integration by the use of partial fractions

Solution:

(a) A=2, B=-5, C=0

(b) 2x/(x2+3)

(c) ln(2x-1) – (5/2)ln(x2+3) + C

(a)6 + 5x – 8x2 = A(x2 + 3) + (Bx + c)(2x – 1)

Sub x = ½

6 + 5/2 – 8(1/2)2 = A(1/4 + 3) + 0

13/2 = (13/4)A —— A=2

When x=0, 6 = 2(3) + c(-1) —— C=0

Compare coefficients of x2

-8 = A + 2B

-10 – 2B —— B = -5

 

(b)d/dx[ln(x2+3)] = 2x/(x2+3)

=2x/(x2+3)

 

(c)∫ 2/(2x-1) + ∫(-5x)/(x2+3)

=ln(2x-1) – (5/2)ln(x2+3) + C

 

Integration 1

Differentiate y = 5xe2x+1 with respect to x and hence, find ∫x(e2x+1) dx.

Solution: 1/2(x)(e2x+1) – 1/4(e2x+1) + C

dy/dx = 5x(e2x+1)(2) + 5(e2x+1)

          =10x(e2x+1) + 5(e2x+1)

5x(e2x+1) = ∫10x(e2x+1) + 5(e2x+1) dx

5x(e2x+1) = 10∫x(e2x+1) dx + 5∫(e2x+1) dx

10∫x(e2x+1) dx + 5/2(e2x+1) = 5x(e2x+1)

10∫x(e2x+1) dx = 5x(e2x+1) -5/2(e2x+1)

∫x(e2x+1) dx = 1/2(x)(e2x+1) – 1/4(e2x+1) + C