Mathematics

Evaluate the following integral:
$$\displaystyle \int { \cfrac { 2x-1 }{ { \left( x-1 \right)  }^{ 2 } }  } dx\quad \quad $$


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

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

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

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

Let $$(x-1)=t\ \Rightarrow\ dx=dt$$

$$=2\displaystyle\int{\dfrac{dx}{\left(t\right)}}+\displaystyle\int{\dfrac{dx}{{\left(t\right)}^{2}}}$$

$$=2\log{\left|t\right|}-\dfrac{1}{t}+c$$           $$\left[\because \displaystyle\int{\dfrac{dx}{{x}^{2}}}=\dfrac{-1}{x},\int{\dfrac{1}{x}}dx=\log |x|\right]$$

$$=2\log{\left|x-1\right|}-\dfrac{1}{x-1}+c$$ where $$t=x-1$$
View Full Answer

Its FREE, you're just one step away


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

Realted Questions

Q1 Single Correct Medium
$$\displaystyle \int \dfrac{xe^x}{(1 + x)^2} dx$$ is equal to
  • A. $$\dfrac{-e^x}{x + 1} + C$$
  • B. $$\dfrac{xe^x}{x + 1} + C$$
  • C. $$\dfrac{-xe^x}{x + 1} + C$$
  • D. $$\dfrac{e^x}{(x + 1)^2} + C$$
  • E. $$\dfrac{e^x}{x + 1} + C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Hard
Solve $$\displaystyle \int {\dfrac{{5x - 1}}{{3{x^2} + x + 2}}} dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 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. $$Ae^ {a}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 One Word Medium
Integral of $$\displaystyle\int  x^{13/2}\left ( 1+x^{5/2} \right )^{1/2}dx$$ can be expressed as $$\displaystyle  \frac{2}{5}\left ( 1+x^{5/2} \right )^{3/2}\left [ \frac{2}{7}\left ( 1+x^{5/2} \right )^{2}-a\left ( 1+x^{b} \right )+\frac{2}{3} \right ]+c$$
then a+(1/b) = ?

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Single Correct Medium
If $$\dfrac{d}{dx}f(x)=g(x)$$ then $$\displaystyle \int_{a}^{b}f(x)g(x)dx=$$
  • A. $$\dfrac{f(b)-f(a)}{2}$$
  • B. $$\dfrac{f(a)-f(b)}{2}$$
  • C. $$\dfrac{{f}^{2}(a)-{f}^{2}(b)}{2}$$
  • D. $$\dfrac{{f}^{2}(b)-{f}^{2}(a)}{2}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer