We can start by factoring the denominator and then using a trigonometric substitution to solve the integral.
First, let's factor the denominator:
1 - 4x - 4x² = -4(x² + x - 1/4) = -4(x + 1/2)(x - 1/2)
Next, we can make the substitution x = (1/2)sin(t) to simplify the integral.
dx = (1/2)cos(t) dt
With these substitutions, the integral becomes:
∫(6(1/2)sin(t) + 5)/(√-4(sin²(t) + sin(t) + 1/4))) (1/2)cos(t) dt
Simplifying:
∫(3sin(t) + 5)/(cos(t)√-4(sin(t) + 1/2) (sin(t) - 1/2)) dt
Now we can use the substitution:
u = sin(t) + 1/2, du = cos(t) dt
With this substitution, the integral becomes:
∫(3u - 7)/(√-4u(u-1)) du
We can simplify this using partial fractions.
(3u - 7)/((u-1)√-4u(u-1)) = A/(u-1) + B/√-4u + C/√u
Multiplying both sides by the denominator and solving for A, B, and C, we get:
A = -1/2, B = 3/2, C = 1
Now the integral becomes:
∫(-1/2)/(u-1) + (3/2)/√-4u + (1)/√u du
Integrating each term:
-1/2 ln|u-1| - (3/2) ln|√-4u| + 2√u + C
Substituting back for u:
-1/2 ln|sin(t) - 1/2| - (3/2) ln|√-2(sin(t) + 1/2)| + 2√(sin(t) + 1/2) + C
Finally, we have our antiderivative in terms of t.
