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)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(x2+2x)32+C\frac{\left(x^2+2x\right)^{^3}}{2}+C−(x2+2x)−1+C-\left(x^2+2x\right)^{^{-1}}+C单项选择题
A
( 𝑥 2 + 2 𝑥 ) 3 6 + 𝐶
B
( 𝑥 2 + 2 𝑥 ) 3 3 + 𝐶
C
− 1 2 ( 𝑥 2 + 2 𝑥 ) − 1 + 𝐶
D
( 𝑥 2 + 2 𝑥 ) 3 2 + 𝐶
E
− ( 𝑥 2 + 2 𝑥 ) − 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)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(x2+2x)32+C\frac{\left(x^2+2x\right)^{^3}}{2}+C−(x2+2x)−1+C-\left(x^2+2x\right)^{^{-1}}+C单项选择题

A

( 𝑥 2 + 2 𝑥 ) 3 6 + 𝐶

B

( 𝑥 2 + 2 𝑥 ) 3 3 + 𝐶

C

− 1 2 ( 𝑥 2 + 2 𝑥 ) − 1 + 𝐶

D

( 𝑥 2 + 2 𝑥 ) 3 2 + 𝐶

E

− ( 𝑥 2 + 2 𝑥 ) − 1 + 𝐶

BlackTom AI · Accepted Answer

We are evaluating multiple choices for the antiderivative of a function that appears to be (x+1)(x^2+2x)^2, based on the structure of the options.
Option 1: ((x^2+2x)^3)/6 + C. If we differentiate this, we get (1/6)*3*(x^2+2x)^2*(2x+2) = (x^2+2x)^2*(x+1). This exactly matches the integrand (x+1)(x^2+2x)^2, so this option is a correct antiderivative.
Option 2: ((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 is twice the desired integrand, so this is incorrect.
Option 3: −(1/2)(x^2+2x) − 1 + C. Differentiating yields −(1/2)*(2x+2) = −(x+1). This is missing the (x^2+2x)^2 factor entirely, so it cannot be the antiderivative of (x+1)(x^2+2x)^2. This is incorrect.
Option 4: ((x^2+2x)^3)/2 + C. Differentiating gives (1/2)*3*(x^2+2x)^2*(2x+2) = 3*(x^2+2x)^2*(x+1). This is three times the desired integrand, so this is incorrect.
Option 5: −(x^2+2x) − 1 + C. Differentiating yields −(2x+2) = −2(x+1). This again lacks the (x^2+2x)^2 factor and does not match the integrand. This is incorrect.

In summary, Option 1 yields the correct antiderivative by straightforward differentiation, while Options 2–5 fail because their derivatives do not reproduce the original integrand or reproduce it with an incorrect factor. The correct match is the first option, ((x^2+2x)^3)/6 + C.