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Question

(4x1)dx2x26x+18 is equal to

A
22x26x+1852ln((x32)+x23x+9+C
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B
22x26x+18+52ln((x32)+x23x+9+C
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C
22x26x+1852ln((x32)+x23x+9+C
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D
22x26x+18+52ln((x32)+x23x+9+C
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Solution

The correct option is D 22x26x+18+52ln((x32)+x23x+9+C
Solving integrals of these types recquires expressing the numerator of the integrand as the sum of derivative of the denominator and a constant.
So here, we can express the numerator as:
(4x1)=addx(2x26x+18)+b
(4x1)=a(4x6)+b
Thus, comparing the coefficient of x and constant term we get
a=1, b=5
Now ,we can write the integral as:
I=(4x1)dx2x26x+18I=(4x6)+52x26x+18dxI=(4x6)2x26x+18dx +5dx2x26x+18I=22x26x+18+I1
Now, we can evaluate I1 as:
I1=5dx2x26x+18I1=52dxx23x+9I1=52dxx23x+9494+9I1=52dx(x32)2274I1=52dx(x32)2(272)2Now, using the formula dxx2a2=ln(x+x2a2+C,we getI1=52ln((x32)+x23x+9 +C
Thus the integral is
I=22x26x+18+ 52ln((x32)+x23x+9 +C
Thus, Option d. is correct.

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