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

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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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