What is the definite integral from 0 to x of 1/(4 + t + 3t^2) dt? For starters, what is the anti-derivative and how did you find it? I think I could take it from there.

To find the anti-derivative of the given function, we need to determine which function, when differentiated, yields 1/(4 + t + 3t^2).

We start by observing that the expression in the denominator resembles a quadratic expression. We can factorize it as follows: 3t^2 + t + 4. However, attempting to find the derivative of this expression would be quite complex.

To simplify the process, we use a technique called "partial fraction decomposition." The idea is to rewrite the integrand as a sum of simpler fractions, which can then be more easily integrated.

To decompose the expression, we assume that it can be expressed as the sum of two fractions: A/(t - r1) and B/(t - r2), where r1 and r2 are the roots of the denominator polynomial (3t^2 + t + 4).

Thus, we have 1/(4 + t + 3t^2) = A/(t - r1) + B/(t - r2)

To find A and B, we multiply through by the denominator and equate the numerators:

1 = A(t - r2) + B(t - r1)

Expanding and combining like terms, we have:

1 = (A + B)t - (Ar2 + Br1) - (Ar1 + Br2)

Since the left side of the equation is just a constant (1), and the right side is a linear polynomial in t, then the coefficients of the terms on both sides must be equal. This gives us a system of equations:

A + B = 0 (coefficient of t^0)
-Ar2 - Br1 = 0 (coefficient of t^1)
-Ar1 - Br2 = 1 (coefficient of t^0)

Solving this system of equations will give us the values of A and B.

Once we find these values, we can rewrite the integrand as:

1/(4 + t + 3t^2) = A/(t - r1) + B/(t - r2)

Finally, we integrate each term individually. The integral of A/(t - r1) is A ln|t - r1|, and the integral of B/(t - r2) is B ln|t - r2|.

Therefore, the antiderivative of 1/(4 + t + 3t^2) with respect to t is:

A ln|t - r1| + B ln|t - r2|

We found the antiderivative using partial fraction decomposition and then integrating each term. You can now proceed with finding the definite integral from 0 to x by substituting the limits into the antiderivative expression and evaluating it.