text: Mathematics
  
  # $\int { P\left( x \right) { e }^{ kx }dx=Q\left( x \right) { e }^{ 4x }+C }$, where $P(x)$ is polynomial of degree $n$ and $Q(x)$ is polynomial of degree $7$. Then the value of $n+7+k+\lim _{ x\rightarrow \infty }{ \dfrac { P\left( x \right) }{ Q\left( x \right) } }$ is:
  
  $22$
  
  Its FREE, you're just one step away
  
  Single Correct Medium Published on 17th 09, 2020
  Questions 203525
  Subjects 9
  Chapters 126
  Enrolled Students 84
  
  #### Realted Questions
  
  Q1 Subjective Medium
  Evaluate the following $\displaystyle \underset{1}{\overset{2}{\int}} \dfrac{5x^2}{x^2 + 4x + 3}$
  
  1 Verified Answer | Published on 17th 09, 2020
  
  Q2 Single Correct Medium
  Let $\displaystyle\int _{ 0 }^{ 1 }{ \dfrac { { e }^{ t }dt }{ 1+t } }$ then $\displaystyle \int _{ a-1 }^{ a }{ \dfrac { { e }^{ t }dt }{ t-a-1 } }$
  • A. $Ae^ {-a}$
  • B. $-Ae^ {-a}$
  • C. None of these
  • D. $Ae^ {a}$
  
  1 Verified Answer | Published on 17th 09, 2020
  
  Q3 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
  
  Q4 Single Correct Hard
  $\displaystyle\int{\frac{{(x+a)}^3}{x^3}+\frac{1-x^4}{1-x}-\frac{7}{x\sqrt{x^2-1}}+\frac{x+2}{{(x+1)}^2}+\frac{{(1+x)}^3}{\sqrt{x}}dx}$ is equal to
  • A. $\displaystyle x+2a\log{x}-\frac{3a^2}{x}-\frac{a^3}{2x^2}+x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}-7\sec^{-1}{x}+\log{(x+1)}\\ -\frac{1}{x+1} +2\sqrt{x}+2x^{\displaystyle\frac{3}{2}}+\frac{6}{5}x^{\displaystyle\frac{5}{2}}+\frac{2}{7}x^{\displaystyle\frac{7}{2}}+C$
  • B. $\displaystyle x+3a\log{x}-\frac{3a^2}{x}-\frac{a^3}{2x^2}+x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}-7\sec^{-1}{x}+\log{(x+1)}\\ -\frac{1}{x+1} +2\sqrt{x}+2x^{\displaystyle\frac{3}{2}}+\frac{6}{5}x^{\displaystyle\frac{5}{2}}+\frac{2}{7}x^{\displaystyle\frac{5}{2}}+C$
  • C. none of these
  • D. $\displaystyle x+3a\log{x}-\frac{3a^2}{x}-\frac{a^3}{2x^2}+x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}-7\sec^{-1}{x}+\log{(x+1)}\\ -\frac{1}{x+1} +2\sqrt{x}+2x^{\displaystyle\frac{3}{2}}+\frac{6}{5}x^{\displaystyle\frac{5}{2}}+\frac{2}{7}x^{\displaystyle\frac{7}{2}}+C$
  
  Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$
explanation: The given text contains a math question and an answer to the question.
  
  The question is:
  Evaluate the integral $\displaystyle\int { \dfrac{5x^2}{x^2 + 4x + 3} } dx$.
  
  The answer provided is:
  $22$
  
  However, the answer provided does not seem to match the question. The answer $22$ is a constant value, while the integral in the question would result in a function of $x$. Therefore, the answer given does not appear to be correct for the question asked.
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
  
  # $\int { P\left( x \right) { e }^{ kx }dx=Q\left( x \right) { e }^{ 4x }+C }$, where $P(x)$ is polynomial of degree $n$ and $Q(x)$ is polynomial of degree $7$. Then the value of $n+7+k+\lim _{ x\rightarrow \infty }{ \dfrac { P\left( x \right) }{ Q\left( x \right) } }$ is:
  
  $22$
  
  Its FREE, you're just one step away
  
  Single Correct Medium Published on 17th 09, 2020
  Questions 203525
  Subjects 9
  Chapters 126
  Enrolled Students 84
  
  #### Realted Questions
  
  Q1 Subjective Medium
  Evaluate the following $\displaystyle \underset{1}{\overset{2}{\int}} \dfrac{5x^2}{x^2 + 4x + 3}$
  
  1 Verified Answer | Published on 17th 09, 2020
  
  Q2 Single Correct Medium
  Let $\displaystyle\int _{ 0 }^{ 1 }{ \dfrac { { e }^{ t }dt }{ 1+t } }$ then $\displaystyle \int _{ a-1 }^{ a }{ \dfrac { { e }^{ t }dt }{ t-a-1 } }$
  • A. $Ae^ {-a}$
  • B. $-Ae^ {-a}$
  • C. None of these
  • D. $Ae^ {a}$
  
  1 Verified Answer | Published on 17th 09, 2020
  
  Q3 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
  
  Q4 Single Correct Hard
  $\displaystyle\int{\frac{{(x+a)}^3}{x^3}+\frac{1-x^4}{1-x}-\frac{7}{x\sqrt{x^2-1}}+\frac{x+2}{{(x+1)}^2}+\frac{{(1+x)}^3}{\sqrt{x}}dx}$ is equal to
  • A. $\displaystyle x+2a\log{x}-\frac{3a^2}{x}-\frac{a^3}{2x^2}+x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}-7\sec^{-1}{x}+\log{(x+1)}\\ -\frac{1}{x+1} +2\sqrt{x}+2x^{\displaystyle\frac{3}{2}}+\frac{6}{5}x^{\displaystyle\frac{5}{2}}+\frac{2}{7}x^{\displaystyle\frac{7}{2}}+C$
  • B. $\displaystyle x+3a\log{x}-\frac{3a^2}{x}-\frac{a^3}{2x^2}+x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}-7\sec^{-1}{x}+\log{(x+1)}\\ -\frac{1}{x+1} +2\sqrt{x}+2x^{\displaystyle\frac{3}{2}}+\frac{6}{5}x^{\displaystyle\frac{5}{2}}+\frac{2}{7}x^{\displaystyle\frac{5}{2}}+C$
  • C. none of these
  • D. $\displaystyle x+3a\log{x}-\frac{3a^2}{x}-\frac{a^3}{2x^2}+x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}-7\sec^{-1}{x}+\log{(x+1)}\\ -\frac{1}{x+1} +2\sqrt{x}+2x^{\displaystyle\frac{3}{2}}+\frac{6}{5}x^{\displaystyle\frac{5}{2}}+\frac{2}{7}x^{\displaystyle\frac{7}{2}}+C$
  
  Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The given text contains a math question and an answer to the question.
  
  The question is:
  Evaluate the integral $\displaystyle\int { \dfrac{5x^2}{x^2 + 4x + 3} } dx$.
  
  The answer provided is:
  $22$
  
  However, the answer provided does not seem to match the question. The answer $22$ is a constant value, while the integral in the question would result in a function of $x$. Therefore, the answer given does not appear to be correct for the question asked.
  
  Label: question-and-answer