Mathematics

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


ANSWER

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


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

Its FREE, you're just one step away


Single Correct Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84
Enroll Now For FREE

Realted Questions

Q1 Subjective Medium
Evaluate $$\displaystyle \int_{0}^{\pi/2} cos \,x \,e^{sin \,x} \,dx$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
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. $$\int _{ 0 }^{ \pi }{ f\left( x \right) dx }$$
  • B. $$\pi \int _{ 0 }^{ \pi }{ f\left( x \right) dx }$$
  • C. $$\frac { { 1 } }{ \pi } \int _{ 0 }^{ \pi }{ f\left( x \right) dx }$$
  • D. $$\frac { { 2 } }{ \pi } \int _{ 0 }^{ \pi }{ f\left( x \right)dx }$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
Evaluate the following integral:
$$\displaystyle\int^{\pi/2}_0\dfrac{\sin x\cos x}{(1+\sin^4x)}dx$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Medium
Evaluate:$$\displaystyle\int \frac{cos\,x}{\sqrt{9sin^{2}x-1}}dx $$
  • A. $$ \log\left | 3 \sin\,x+\sqrt{9 \sin^{2}x-1} \right |+C $$
  • B. $${3} \log\left | 3 \sin\,x+\sqrt{9 \sin^{2}x-1} \right |+C $$
  • C. none of these
  • D. $$\dfrac{1}{3} \log\left | 3 \sin\,x+\sqrt{9 \sin^{2}x-1} \right |+C $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Medium
For the next two (02) items that follow :
Consider the integrals $$I_1=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{dx}{1+\sqrt{tan x}}$$ and $$I_2=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{sin x}dx}{\sqrt{sin }x+\sqrt{cos}x}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer