Mathematics

Solve $\displaystyle\int {\dfrac{{2 + x + {x^2}}}{{{x^2}\left( {2 + x} \right)}} + \dfrac{{2x - 1}}{{{{\left( {x + 1} \right)}^2}}}dx}$

$\ln \left| {2 + x} \right| + \dfrac{3}{{\left( {x + 1} \right)}} + 2\ln \left( {x + 1} \right) - \dfrac{1}{x} + C$

SOLUTION
$\begin{array}{l}\displaystyle\int {\dfrac{{2 + x + {x^2}}}{{{x^2}\left( {2 + x} \right)}} + \dfrac{{2x - 1}}{{{{\left( {x + 1} \right)}^2}}}dx} \\ = \displaystyle\int {\dfrac{{2 + x}}{{\left( {2 + x} \right){x^2}}} + \dfrac{{{x^2}}}{{{x^2}\left( {2 + x} \right)}} + \dfrac{{2x + 2}}{{{{\left( {x + 1} \right)}^2}}} - \dfrac{3}{{{{\left( {x + 1} \right)}^2}}}dx} \\ = \displaystyle\int {\dfrac{1}{{{x^2}}} + \dfrac{1}{{2 + x}}} + \dfrac{{2\left( {x + 1} \right)}}{{{{\left( {x + 1} \right)}^2}}} - \dfrac{3}{{{{\left( {x + 1} \right)}^2}}}dx\\ = - \dfrac{1}{x} + \ln \left| {2 + x} \right| + \dfrac{3}{{\left( {x + 1} \right)}} + \displaystyle\int {\dfrac{{2\left( {x + 1} \right)}}{{{{\left( {x + 1} \right)}^2}}}dx} \end{array}$
Let
$\begin{array}{l}{\left( {x + 1} \right)^2} = t\\2\left( {x + 1} \right)dx = dt\\ = - \dfrac{1}{x} + \ln \left| {2 + x} \right| + \dfrac{3}{{\left( {x + 1} \right)}} + \displaystyle\int {\dfrac{2}{t}dt} \\ = - \dfrac{1}{x} + \ln \left| {2 + x} \right| + \dfrac{3}{{\left( {x + 1} \right)}} + \ln {\left( {x + 1} \right)^2} + C\\ = \ln \left| {2 + x} \right| + \dfrac{3}{{\left( {x + 1} \right)}} + 2\ln \left( {x + 1} \right) - \dfrac{1}{x} + C\end{array}$

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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 $\displaystyle \int_{0}^{\pi/2} cos \,x \,e^{sin \,x} \,dx$.

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
Let $f\left( x \right) =\frac { \sin { x } }{ x }$, then $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ f\left( x \right) f\left( \frac { \pi }{ 2 } -x \right) } dx=$
• A. $\int _{ 0 }^{ \pi }{ f\left( x \right) dx }$
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• D. $\frac { { 2 } }{ \pi } \int _{ 0 }^{ \pi }{ f\left( x \right)dx }$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate the following integral:
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1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Evaluate:$\displaystyle\int \frac{cos\,x}{\sqrt{9sin^{2}x-1}}dx$
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• C. none of these
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Consider the integrals $I_1=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{dx}{1+\sqrt{tan x}}$ and $I_2=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{sin x}dx}{\sqrt{sin }x+\sqrt{cos}x}$