Final answer:
The student's question involves the evaluation of an integral using integration by parts. We identify the parts of the function to differentiate and integrate, then apply the integration by parts formula to solve.
Step-by-step explanation:
The question asks us to evaluate the integral ∫3(2x+1)ln(x²+3)dx. This requires the use of integration techniques. We can approach this by using integration by parts, where we let u = ln(x²+3) and dv = (2x+1)dx. The derivative of u (du) is −2x/(x²+3)dx, and the integral of dv (v) is x² + x. Applying the integration by parts formula, ∫ u dv = uv - ∫ v du, we obtain:
(ln(x²+3))·(x² + x) - ∫(x² + x)·(2x/(x²+3))dx
Now the integral becomes: 3 ∫ln(u)d(2x+1). Integrate ln(u) with respect to u: 3 ∫ln(u)du = 3(u ln(u) - u) + C, where C is the constant of integration.
Substituting u back in terms of x, we have: 3(x^2+3) ln(x^2+3) - 3(x^2+3) + C, where C is the constant of integration.