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Consider int(x^(3)+3x^(2)+2x+1)/(sqrt(...

Consider
`int(x^(3)+3x^(2)+2x+1)/(sqrt(x^(2)+x+1))dx`
`=(ax^(2)+bx+c)sqrt(x^(2)+x+1)+lambda int(dx)/(sqrt(x^(2)+x+1))`
Now, match the following lists and then choose the correct code.

Codes:
`{:(,a,b,c,d),((1),q,p,s,r),((2),s,p,q,r),((3),r,q,p,s),((4),q,s,p,r):}`

Text Solution

Verified by Experts

The correct Answer is:
4
` int(x^(3)+3x^(2)+2x+1)/(sqrt(x^(2)+x+1))dx`
`=(ax^(2)+bx+c)sqrt(x^(2)+x+1)+lambda int(dx)/(sqrt(x^(2)+x+1))`
Differentiating both side, we get
`(x^(3)+3x^(2)+2x+1)/(sqrt(x^(2)+x+1))=((ax^(2)+bx+c)(2x+1))/(2sqrt(x^(2)+x+1))+(2ax+b)sqrt(x^(2)+x+1)+lambda(1)/(sqrt(x^(2)+x+1))`
`=(6ax^(3)+(5a+4b)x^(2)+(4a+3b+2c)x+(c+2b+2lambda))/(2sqrt(x^(2)+x+1))`
Now, compare coefficients on both sides and solve equations.
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