Question at position 1 ∫x+1(x2+2x)2dx=\int\frac{x+1}{\left(x^2+2x\right)^{^2}}dx=(x2+2x)36+C\frac{\left(x^2+2x\right)^{^3}}{6}+C(x2+2x)32+C\frac{\left(x^2+2x\right)^{^3}}{2}+C−(x2+2x)−1+C-\left(x^2+2x\right)^{^{-1}}+C(x2+2x)33+C\frac{\left(x^2+2x\right)^{^3}}{3}+C−12(x2+2x)−1+C-\frac{1}{2}\left(x^2+2x\right)^{^{-1}}+C

Question

Question at position 1 ∫x+1(x2+2x)2dx=\int\frac{x+1}{\left(x^2+2x\right)^{^2}}dx=(x2+2x)36+C\frac{\left(x^2+2x\right)^{^3}}{6}+C(x2+2x)32+C\frac{\left(x^2+2x\right)^{^3}}{2}+C−(x2+2x)−1+C-\left(x^2+2x\right)^{^{-1}}+C(x2+2x)33+C\frac{\left(x^2+2x\right)^{^3}}{3}+C−12(x2+2x)−1+C-\frac{1}{2}\left(x^2+2x\right)^{^{-1}}+C

BlackTom AI · Accepted Answer

We are evaluating the integral ∫ (x+1) / (x^2 + 2x)^2 dx, and we have several candidate antiderivatives to assess.

Option 1: ((x^2 + 2x)^3) / 6 + C
This would differentiate to (1/6) * 3 * (x^2 + 2x)^2 * (2x + 2) by the chain rule, which simplifies to (x^2 + 2x)^2 * (x + 1). This is not equal to (x+1) / (x^2 + 2x)^2, because the derivative yields a factor of (x^2 + 2x)^2 in the numerator rather than the reciprocal of (x^2 + 2x)^2. Therefore, this option is incorrect.

Option 2: ((x^2 + 2x)^3) / 2 + C
Differentiating gives (1/2) * 3 * (x^2 + 2x)^2 * (2x + 2) = (3/2) * (x^2 + 2x)^2 * (x + 1), which again does not produce (x+1) / (x^2 + 2x)^2. The mismatch is even more pronounced than in Option 1, so this is incorrect.

Option 3: − (x^2 + 2x) − 1 + C
Differentiating yields −(2x + 2) = −2(x + 1). This is a polynomial 2nd- or first-degree derivative, not a rational function with (x^2 + 2x)^2 in the denominator. Therefore, this cannot be the antiderivative of (x+1) / (x^2 + 2x)^2, so it is incorrect.

Option 4: ((x^2 + 2x)^3) / 3 + C
Differentiating gives (1/3) * 3 * (x^2 + 2x)^2 * (2x + 2) = 2 * (x^2 + 2x)^2 * (x + 1). This again yields a numerator with (x^2 + 2x)^2, not the reciprocal needed. Hence, incorrect.

Option 5: − (1/2) * (x^2 + 2x)^(−1) + C
Let u = x^2 + 2x. Then du = (2x + 2) dx = 2(x + 1) dx, so (x + 1) dx = du / 2. The integral ∫ (x+1) / (x^2 + 2x)^2 dx becomes ∫ (1 / u^2) * (du / 2) = (1/2) ∫ u^(−2) du = (1/2) * (u^(−1) / (−1)) + C = −1/(2u) + C = −1/[2(x^2 + 2x)] + C, which matches the form of this option (noting that (x^2 + 2x)^(−1) = 1/(x^2 + 2x)). Therefore, this option is correct.

In summary, each option was checked by differentiating or by a standard substitution approach. The correct antiderivative corresponds to the expression with a negative one-half power of (x^2 + 2x), i.e., −1/2 * (x^2 + 2x)^(−1) + C, which aligns with Option 5.

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