Mathematics

# If $\displaystyle \frac{3x^3-8x^2+10}{(x-1)^4} = \frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C}{(x-1)^3} + \frac{D}{(x-1)^4}$, then $A+B+C+D$ is equal to

2

##### SOLUTION
$\dfrac { 3x^{ 3 }-8x^{ 2 }+10 }{ (x-1)^{ 4 } } =\dfrac { A }{ x-1 } +\dfrac { B }{ (x-1)^{ 2 } } +\dfrac { C }{ (x-1)^{ 3 } } +\dfrac { D }{ (x-1)^{ 4 } }$

Multiply both sides by $(x-1)^{ 4 }$

$3x^{ 3 }-8x^{ 2 }+10=A(x-1)^{ 3 }+B(x-1)^{ 2 }+C(x-1)+D$

Expand and collect in terms of power of $x$

$3x^{ 3 }-8x^{ 2 }+10=Ax^{ 3 }+\left( B-3A \right) { x }^{ 2 }+\left( 3A-2B+C \right) x-A+B-C+D$

Gives

$A=3$

$B-3A=-8\Rightarrow B=-8+3(3)=1$

$3A-2B+C=0\Rightarrow C=2B-3A=2-9=-7$

$-A+B-C+D=10\Rightarrow D=10+C-B+A=10-7-1+3=5$

$\therefore A=3,B=1,C=-7,D=5$

Hence $A+B+C+D=2$

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One Word Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Subjective Medium
$\int sin^{2/3}x cos^{3}x dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
If $\displaystyle I = \int \frac {dx}{(2 \sin x + \sec x)^4}$, then I equals
• A. $\displaystyle \frac {1}{5 \tan^5 x} + \frac {1}{3 \tan^6 x} - \frac {I}{(2 \sin x + \sec x)^3} + C$
• B. $\displaystyle \frac {-1}{3(2 \sin x + \sec)^3} + \tan^{-1} (3\sqrt {\tan x}) + C$
• C. $\displaystyle \frac {-1}{3(2 \sin x + \sec x)^3} - \tan^{-1} (3\sqrt {\tan x}) + C$
• D. $\displaystyle -\frac {1}{5 \tan^5 x} + \frac {1}{3 \tan^6 x} - \frac {2}{7 \tan^7 x} + C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
The value of $\displaystyle\int { \dfrac { dx }{ \sqrt { 2x-{ x }^{ 2 } } } }$ is
• A. $\sin ^{ -1 }{ \left( x \right) } +c$
• B. $\sin ^{ -1 }{ \left( 1+x \right) } +c$
• C. $-\sqrt { 2x-{ x }^{ 2 } } +c$
• D. $\sin ^{ -1 }{ \left( x-1 \right) } +c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Hard

$\displaystyle\int {\frac{{2x - 3}}{{\left( {{x^2} - 1} \right)\left( {2x + 3} \right)}}dx}$

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