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:

$22$

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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 $\displaystyle f(x)>0 \forall x \in R$