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Question

If∫x+8x2+6x+5dx=a ln(x2+6x+5∣∣+b ln(x+1x+5∣∣+Cthen the value of ab will be

A
516
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B
58
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C
54
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D
52
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Solution

The correct option is B 58Solving these types of equations requires expressing numerator in terms of derivative of the denominator of the integrand and a constant value, i.e. for integrand of the form ND we write N=cddx(D)+d, where c and d are constants here, we can write, x+8=cddx(x2+6x+5)+d⇒x+8=c(2x+6)+d⇒x+8=2cx+(6c+d) Now, comparing the coefficients of x and constant value we get, c=12, d=5 now, we can write the integral as I=∫x+8x2+6x+5dx⇒I=∫12(2x+6)+5x2+6x+5⇒I=12∫2x+6x2+6x+5dx+5∫dxx2+6x+5⇒I=12ln(x2+6x+5∣∣ +5∫dx(x+1)(x+5) Now, we can solve the second integral by integration by partial fraction. So, we get the integral as I=12ln(x2+6x+5∣∣ +54ln(x+1x+5∣∣+C Now, comparing we get, a=12, b=54thus ab=58 Thus, Option b. is correct.

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