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Find the definite integral of f(x) = 12/(x^2+10x+21) with limits [-1,1]. Give your answer to 2 decimal places.
Factorising denominator of fraction:
x^2 + 10x + 21 = (x+3)(x+7)
2)
Partial fractions:
f(x) = 12/(x+3)(x+7)
; let
f(x) = A/(x+3) + B/(x+7)
then equating the nominator:
A(x+7) + B(x+3) = 12
From this we can set up simultaneous equations: (1):
Ax + Bx = 0
; (2):
7A + 3B = 12
From (1):
A = -B
Substituting A into (2):
7(-B) + 3B =12
So
B=-3
, and
A=3f(x) = 3/(x+3) – 3/(x+7)
3)
Integrating the function:
∫ f(x) dx = 3 ∫ (1/(x+3) - 1/(x+7)) dx = 3ln|x+3| - 3ln|x+7| + c
4)
Evaluating the integral with limits [-1,1]:
3ln|1+3| - 3ln|1+7| - 3ln|-1+3| + 3ln|-1+7| = 3ln(3/2) = ln(27/8) =
1.22 (2d.p.)
CG
Answered by
Camilla Giulia B.
•
Further Mathematics tutor
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