Replace the ? with a function of x that will make the integrand equal to the derivative of a product, and then find the
antiderivative. Choose 0 for the constant of integration.
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Question

Quizplus · Accepted Answer

To make the integrand equal to the derivative of a product, we can use the product rule of differentiation. Let's choose the function x^2 as the first term of the product and ? as the second term. Then we have:

x^2 * ?' + 2x * ? = (2x + 1) / (x^2 + x)

Now, we can see that the left-hand side is the derivative of the product x^2 * ?, so we need to solve for ?'. Rearranging the terms, we get:

?' = [(2x + 1) / (x^2 + x) - 2x * ?] / x^2

Now we can integrate both sides with respect to x (using partial fractions on the right-hand side), and we get:

? = (-1/2) * ln(x^2 + x) + (3/2) * ln(x) + C

where C is the constant of integration. Since we are told to choose 0 for the constant of integration, we get:

? = (-1/2) * ln(x^2 + x) + (3/2) * ln(x)

Replace the ? with a Function of X That Will