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) = 12From this we can set up simultaneous equations: (1): Ax + Bx = 0; (2): 7A + 3B = 12From (1): A = -BSubstituting A into (2): 7(-B) + 3B =12So 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| + c4) 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.)
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