text: Mathematics
  
  # Evaluate:$\displaystyle\int{\dfrac{{x}^{2}}{1+{x}^{3}}}dx$
  
  ##### SOLUTION
  $\begin{array}{l}\text { Let } 1+x^{3}=z \\\quad \Rightarrow 3 x^{2} d x=d z \quad \Rightarrow x^{2} \mathrm{d} x=\dfrac{d z}{3} \\\therefore \quad \displaystyle \int \dfrac{x^{2}}{1+x^{3}} \mathrm{d} \mathbf{x}\\=\dfrac{1}{3} \displaystyle \int \dfrac{\mathrm{d} z}{z}\\=\dfrac{1}{3} \log |z|+C\\=\dfrac{1}{3} \log \left|1+x^{3}\right|+C\end{array}$
  
  Its FREE, you're just one step away
  
  Subjective Medium Published on 17th 09, 2020
  Questions 203525
  Subjects 9
  Chapters 126
  Enrolled Students 86
  
  #### Realted Questions
  
  Q1 Subjective Medium
  Evaluate the integrals
  $\int _{ 0 }^{ 2 }{ \cfrac { 1 }{ \left( { x }^{ 2 }+1 \right) } dx }$
  
  1 Verified Answer | Published on 17th 09, 2020
  
  Q2 Single Correct Hard
  Let $\displaystyle f(x)=\int{\frac{x^2}{(1+x^2)(1+\sqrt{1+x^2})}dx}$ and $f(0)=0$. Then $f(1)$ is equal to
  • A. $\log_e{(1+\sqrt{2})}$
  • B. $\displaystyle\log_e{(1+\sqrt{2})}+\frac{\pi}{4}$
  • C. none of these
  • D. $\displaystyle\log_e{(1+\sqrt{2})}-\frac{\pi}{4}$
  
  1 Verified Answer | Published on 17th 09, 2020
  
  Q3 One Word Medium
  Evaluate:$\displaystyle\int \frac{dx}{\sqrt{4x^{2}-1}}$
  
  1 Verified Answer | Published on 17th 09, 2020
  
  Q4 Single Correct Medium
  $\displaystyle\int_{0}^{\pi}\frac{dx}{1+a^{\cos x}}$ equals
  
  • A. $0$
  • B. $\displaystyle \pi$
  • C. None of these
  • D. $\displaystyle\frac{\pi }{2}$
  
  1 Verified Answer | Published on 17th 09, 2020
  
  Q5 Single Correct Medium
  $\int { (\cfrac { { 2 }^{ x }-5^{ x } }{ 10^{ x } } )dx }$ is equal to __________________.
  • A. $\cfrac { { 2 }^{ x } }{ \log _{ e }{ 2 } } -\cfrac { 5^{ x } }{ \log _{ e }{ 5 } } +c$
  • B. $\cfrac { { 2 }^{ x } }{ \log _{ e }{ 2 } } +\cfrac { 5^{ x } }{ \log _{ e }{ 5 } } +c$
  • C. $\cfrac { { 5 }^{ -x } }{ \log _{ e }{ 5 } } -\cfrac { 2^{ -x } }{ \log _{ e }{ 2 } } +c$
  • D. $\cfrac { { 2 }^{ -x } }{ \log _{ e }{ 2 } } -\cfrac { 5^{ -x } }{ \log _{ e }{ 5 } } +c$
explanation: The given text is a math problem with a solution. It contains a question and an answer.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: Mathematics
  
  # Evaluate:$\displaystyle\int{\dfrac{{x}^{2}}{1+{x}^{3}}}dx$
  
  ##### SOLUTION
  $\begin{array}{l}\text { Let } 1+x^{3}=z \\\quad \Rightarrow 3 x^{2} d x=d z \quad \Rightarrow x^{2} \mathrm{d} x=\dfrac{d z}{3} \\\therefore \quad \displaystyle \int \dfrac{x^{2}}{1+x^{3}} \mathrm{d} \mathbf{x}\\=\dfrac{1}{3} \displaystyle \int \dfrac{\mathrm{d} z}{z}\\=\dfrac{1}{3} \log |z|+C\\=\dfrac{1}{3} \log \left|1+x^{3}\right|+C\end{array}$
  
  Its FREE, you're just one step away
  
  Subjective Medium Published on 17th 09, 2020
  Questions 203525
  Subjects 9
  Chapters 126
  Enrolled Students 86
  
  #### Realted Questions
  
  Q1 Subjective Medium
  Evaluate the integrals
  $\int _{ 0 }^{ 2 }{ \cfrac { 1 }{ \left( { x }^{ 2 }+1 \right) } dx }$
  
  1 Verified Answer | Published on 17th 09, 2020
  
  Q2 Single Correct Hard
  Let $\displaystyle f(x)=\int{\frac{x^2}{(1+x^2)(1+\sqrt{1+x^2})}dx}$ and $f(0)=0$. Then $f(1)$ is equal to
  • A. $\log_e{(1+\sqrt{2})}$
  • B. $\displaystyle\log_e{(1+\sqrt{2})}+\frac{\pi}{4}$
  • C. none of these
  • D. $\displaystyle\log_e{(1+\sqrt{2})}-\frac{\pi}{4}$
  
  1 Verified Answer | Published on 17th 09, 2020
  
  Q3 One Word Medium
  Evaluate:$\displaystyle\int \frac{dx}{\sqrt{4x^{2}-1}}$
  
  1 Verified Answer | Published on 17th 09, 2020
  
  Q4 Single Correct Medium
  $\displaystyle\int_{0}^{\pi}\frac{dx}{1+a^{\cos x}}$ equals
  
  • A. $0$
  • B. $\displaystyle \pi$
  • C. None of these
  • D. $\displaystyle\frac{\pi }{2}$
  
  1 Verified Answer | Published on 17th 09, 2020
  
  Q5 Single Correct Medium
  $\int { (\cfrac { { 2 }^{ x }-5^{ x } }{ 10^{ x } } )dx }$ is equal to __________________.
  • A. $\cfrac { { 2 }^{ x } }{ \log _{ e }{ 2 } } -\cfrac { 5^{ x } }{ \log _{ e }{ 5 } } +c$
  • B. $\cfrac { { 2 }^{ x } }{ \log _{ e }{ 2 } } +\cfrac { 5^{ x } }{ \log _{ e }{ 5 } } +c$
  • C. $\cfrac { { 5 }^{ -x } }{ \log _{ e }{ 5 } } -\cfrac { 2^{ -x } }{ \log _{ e }{ 2 } } +c$
  • D. $\cfrac { { 2 }^{ -x } }{ \log _{ e }{ 2 } } -\cfrac { 5^{ -x } }{ \log _{ e }{ 5 } } +c$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The given text is a math problem with a solution. It contains a question and an answer.
  
  Label: question-and-answer