It is known that int_a^5 x^2+4x+1 dx = 132, what is the value of a.

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gsarora17 eNotes educator| Certified Educator











Since the discriminant of this cubic equation is less than zero, so the equation has one real root and two complex conjugate roots.

Now to solve this cubic equation , let us depress the cubic equation by substituting a with (y-2).

So, `(y-2)^3+6(y-2)^2+3(y-2)+106=0`  




Now we have to solve this depressed cubic equation of the form y^3+Ay=B

`y=s-t , A=3st=-9 , B=s^3-t^3=-116`






Now to solve the above equation for t , let us reduce to the quadratic form by assuming , t^3=u.

So the equation reduces to,`u^2-116u+27=0`




`u=(116+-sqrt(4*3337))/2 = 58+-sqrt(3337)`




`y=s-t and a=y-2`

` :.a=-2+s-t`


` or a=-2-3/root(3)(58-sqrt(3337))-root(3)(58-sqrt(3337)) `

`or a=-7.48927`

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