Mathematics

# Solve $\displaystyle\int {\frac{{{e^x}\left( {{x^2} + 5x + 7} \right)}}{{{{\left( {x + 3} \right)}^2}}}} dx$

##### SOLUTION
$\displaystyle\int {\dfrac{{{e^x}\left( {{x^2} + 5x + 7} \right)}}{{{{\left( {x + 3} \right)}^2}}}} dx$

let
${u^2} = {\left( {x + 3} \right)^2}\\$

$\implies {x^2} + 5x + 7 = {u^2} - u + 1\\$

$\displaystyle \int {\dfrac{{{e^{u - 3}}\left( {{u^2} - u + 1} \right)}}{{{{\left( u \right)}^2}}}} du\\$

$\displaystyle={e^{-3}}\int {{e^u}\left( {\dfrac{1}{{{u^2}}} - \dfrac{1}{u} + 1} \right)du\,} \,$

$\displaystyle= {e^{ - 3}}\int {{e^u}du} \, + {e^{ - 3}}\int {\left( {\dfrac{1}{u} - \dfrac{1}{{{u^2}}}} \right){e^u}du} \,\\$

$\displaystyle= {e^{ - 3}}{e^u} + \dfrac{{{e^{ - 3}}}}{u}$

$= {e^{u - 3}} + \dfrac{{{e^{ - 3}}}}{{u }}$

Putting the value of u, we get

$= {e^x} + \dfrac{{{e^{ - 3}}}}{{x + 3}} + c$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Hard
$\displaystyle\int\, \cos\, 2\theta.\,\ln\, \displaystyle \frac {\cos\theta\, +\, \sin\, \theta}{\cos\, \theta\, -\, \sin\, \theta}\, d\theta$
• A. $\, \displaystyle \frac {1}{2}\, ln\, \left ( \displaystyle \frac {\cos\, \theta\, -\, \sin\, \theta}{\cos\, \theta\, +\, \sin\, \theta} \right )\, \sin\, 2\, \theta\, -\, \displaystyle \frac {1}{2}\, \ln\, (\sec\, 2\,\theta)\, +\, C$
• B. $\, \displaystyle \frac {1}{2}\, \ln\, \left ( \displaystyle \frac {\cos\, \theta\, -\, \sin\, \theta}{\cos\, \theta\, +\, \sin\, \theta} \right )\, \sin\, 2\, \theta\, -\, \displaystyle \frac {1}{2}\, \ln\, (\text{cosec}\, 2\,\theta)\, +\, C$
• C. $\, \displaystyle \frac {1}{2}\, \ln\, \left ( \displaystyle \frac {\cos\, \theta\, +\, \sin\, \theta}{\cos\, \theta\, -\, \sin\, \theta} \right )\, \sin\, 2\, \theta\, -\, \displaystyle \frac {1}{2}\, \ln\, (\text{cosec}\, 2\,\theta)\, +\, C$
• D. $\, \displaystyle \frac {1}{2}\, \ln\, \left ( \displaystyle \frac {\cos\, \theta\, +\, \sin\, \theta}{\cos\, \theta\, -\, \sin\, \theta} \right )\, \sin\, 2\, \theta\, -\, \displaystyle \frac {1}{2}\, \ln\, (\sec\, 2\,\theta)\, +\, C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
For $X \rightarrow B (n,p), n =10, E(X)=5$< then $p=$
• A. $0.1$
• B. $0.05$
• C. $0.5$
• D. $0.01$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\displaystyle\int (2x+1)\sqrt{x^2+x+1}dx$.
• A. $\dfrac{2}{5}(x^2+x+1)^\dfrac{5}{2} + C$
• B. $(x^2+x+1)^\dfrac{3}{2} + C$
• C. $\dfrac{2}{3}(x^2+x+1)^\dfrac{1}{2} + C$
• D. $\dfrac{2}{3}(x^2+x+1)^\dfrac{3}{2} + C$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate the following integral:
$\displaystyle\int_{0}^{2} (3x+2)\ dx$

Let $\displaystyle f\left ( x \right )=\frac{\sin 2x \cdot \sin \left ( \dfrac{\pi }{2}\cos x \right )}{2x-\pi }$