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

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

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.

View Explanation

Log in for full answers

Similar Questions

Evaluate the integral.

Find the integral ∫ 𝑥 2 ( 2 𝑥 3 + 1 ) 4 𝑑 𝑥

∫ sec 2 ⁡ 𝑥 tan 7 ⁡ 𝑥 𝑑 𝑥 = _ _ _ _ _ _ _ _ _ _ _

Find ∫ 2 𝑥 3 𝑑 𝑥

∫9x dx=(coefficient)x(power)+c\displaystyle\int 9\sqrt{x} \ dx=\text{(coefficient)}x^{\text{(power)}} +c where coefficient = [Fill in the blank] and power = [Fill in the blank] (convert any fractional answers to decimal)

Question at position 5 ∫323dx=\int3^{\frac{2}{3}}dx=323x+C3^{\frac{2}{3}}x+C35353+C\frac{3}{5}3^{\frac{5}{3}}+C53353+C\frac{5}{3}3^{\frac{5}{3}}+C000+C0+C

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler