Mathematics

Evaluate $$\displaystyle \int \dfrac{x-3}{(x-1)^{3}} dx$$


SOLUTION
$$\displaystyle \int{\dfrac{\left(x-3\right)dx}{{\left(x-1\right)}^{3}}}$$
$$=\displaystyle \int{\dfrac{\left(x-1-2\right)dx}{{\left(x-1\right)}^{3}}}$$
$$=\displaystyle \int{\dfrac{\left(x-1\right)dx}{{\left(x-1\right)}^{3}}}-2\int{\dfrac{dx}{{\left(x-1\right)}^{3}}}$$
$$=\displaystyle \int{\dfrac{dx}{{\left(x-1\right)}^{2}}}-2\int{\dfrac{dx}{{\left(x-1\right)}^{3}}}$$
$$=\displaystyle \int{{\left(x-1\right)}^{-2}dx}-2\int{{\left(x-1\right)}^{-3}dx}$$
$$=\dfrac{{\left(x-1\right)}^{-2+1}}{-2+1}-2\dfrac{{\left(x-1\right)}^{-3+1}}{-3+1}$$
$$=\dfrac{{\left(x-1\right)}^{-1}}{-1}-2\dfrac{{\left(x-1\right)}^{-2}}{-2}$$
$$=\dfrac{-1}{x-1}+\dfrac{1}{{\left(x-1\right)}^{2}}+c$$
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
The value of $$\displaystyle \int \dfrac{dx}{\sqrt {x} (x + 9)} dx$$ is equal to:
  • A. $$\tan ^{-1} \sqrt{x} + c$$
  • B. $$\tan ^{-1} (\dfrac{\sqrt{x}}{3}) + c$$
  • C. $$\dfrac{2}{3} \tan ^{-1} \sqrt{x} + c$$
  • D. $$\dfrac{2}{3} \tan ^{-1} (\dfrac{\sqrt{x}}{3}) + c$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Hard
Evaluate the following integral:
$$\int { \cfrac { { x }^{ 2 }+x+1 }{ { x }^{ 2 }-x }  } dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Hard
If $$\displaystyle I=\frac{\cot x}{\sqrt{a+b\cot^{2}x}}dx\left ( 0< a < b \right ),$$ then I equals

  • A. $$\displaystyle \sqrt{b-a}

    \sin ^{-1}\left ( \sqrt{b-a}x \right )+Const$$
  • B. $$\displaystyle \sqrt{b-a}\sin ^{-1}\left ( \sqrt{b-a}\sin x \right )+Const$$
  • C. $$-{\sqrt{b-a}\cos^{-1}}\left ( \displaystyle \sqrt{\frac{b-a}{b}\sin x} \right )+Const$$
  • D. $$\displaystyle \frac{1}{\sqrt{b-a}}\sin ^{-1}\left ( \sqrt{\frac{b-a}{b}\sin x} \right )+Const$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Hard
Evaluate:
$$\int { \sqrt { 9+{ x }^{ 2 } }  } dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Medium

Evaluate the  integrals :

$$\int\limits_1^2 {\dfrac{{{{\left( {x + 1} \right)}^2}}}{{\sqrt x }}dx} $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer