Final answer:
The method of substitution reduces the integral to ∠ ln(u) du. With u = 2x² + 3 and du = 4x dx, the integral becomes (2/3) * ∠ ln(u) du which integrates to (2/3) * (u ln(u) - u), then substituting back for u gives the antiderivative in terms of x.
Step-by-step explanation:
The integral given is ∠ (8x ln(2x² +3)) / (6x² +9) dx. First, notice that the denominator 6x² + 9 can be rewritten as 3*(2x² + 3). This suggests a substitution: Let u = 2x² + 3. Then du = 4x dx, and we notice that 8x dx can be written as 2 * 4x dx, which will help us use the du substitution.
To find the correct expression for dx, we solve the equation du = 4x dx for dx: dx = du/(4x). Now we can substitute u and dx for their equivalents in terms of x, yielding the new integral: ∠ ln(u) * 2 * du/3. Factor the constant out of the integral to get (2/3) * ∠ ln(u) du. Now, we can integrate ln(u) with respect to u.
The integral of ln(u) is u ln(u) - u. So, the antiderivative is (2/3) * (u ln(u) - u). Substituting back in for u, we get (2/3) * ((2x² + 3) ln(2x² + 3) - (2x² + 3)).
This result represents the indefinite integral of the original expression. To simplify further, we can distribute the (2/3) and combine like terms if appropriate, resulting in a simplified version of the antiderivative.