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118. $\frac{x}{{(x - 1)}^{2} (x - 2)}$

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Vishal Baghel | Contributor-Level 10

4 months ago

$\frac{x}{{(x - 1)}^{2} (x - 2)} = \frac{A}{(x - 1)} + \frac{B}{{(x - 1)}^{2}} + \frac{C}{x + 2}$

= A(x- 1)(x + 2) + B(x + 2) + C(x- 1)²

= A(x² + 2x-x- 2) + B(x + 2) + C(x² + 1 - 2x)

= A(x² + x- 2) + B(x + 2) + C(x²- 2x + 1)

Comparing the co-efficient we get, A + C = 0 ¾ (1)

A + B - 2C = 1 ¾ (2)

- 2A + 2B + C = 0 ¾ (3)

EQⁿ (3) - 2 ´ eⁿQ. (2),

- 2A + 2B + C (2A + 2B - 4C) = 0 - 2 ´ 1.

=> - 4A + 5C = - 2 ¾ (4)

Eqⁿ (4) + 4 ´ Eqⁿ (1) we get,

- 4A + 5C + 4A + 4C = - 2 + 4 ´ 0

=> 9C = - 2

$\begin{array}{l} \Rightarrow C = - \frac{2}{9} \\ P u t t i n g C = - \frac{2}{9}, i n - - - - - (1) \\ P u t t i n g v a l u e o f A a n d C i n (3) w e g e t, \\ - 2 \times \frac{2}{9} + 2 B - \frac{2}{9} = 0 \\ \Rightarrow 2 B = \frac{2}{9} + \frac{4}{9} \\ \Rightarrow 2 B = \frac{6}{9} : \frac{2}{3} \\ \Rightarrow B = \frac{1}{3} . \end{array}$

<math><mrow><mfrac><mrow><mi>x</mi></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>A</mi></mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>B</mi></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>C</mi></mrow><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></math>= A(x- 1)(x + 2) + B(x + 2) + C(x- 1)2= A(x2 + 2x-x- 2) + B(x + 2) + C(x2 + 1 - 2x)= A(x2 + x- 2) + B(x + 2) + C(x2- 2x + 1)Comparing the co-efficient we get, A + C = 0 ¾ (1)A + B - 2C = 1 ¾ (2)- 2A + 2B + C = 0 ¾ (3)EQn (3) - 2 ´ enQ. (2),- 2A + 2B + C (2A + 2B - 4C) = 0 - 2 ´ 1.=> - 4A + 5C = - 2 ¾ (4)Eqn (4) + 4 ´ Eqn (1) we get,- 4A + 5C + 4A + 4C = - 2 + 4 ´ 0=> 9C = - 2 <math> <mtable columnalign="left"> <mtr> <mtd> <mrow> <mo>⇒</mo> <mi>C</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>9</mn> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow><mi>P</mi><mi>u</mi><mi>t</mi><mi>t</mi><mi>i</mi><mi>n</mi><mi>g</mi><mtext> </mtext><mi>C</mi><mo>=</mo><mo>−</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>9</mn></mrow></mfrac><mo>,</mo><mo> </mo><mi>i</mi><mi>n</mi><mo>−</mo><mo>−</mo><mo>−</mo><mo>−</mo><mo>−</mo> <mrow><mo>(</mo><mrow><mn>1</mn></mrow><mo>)</mo></mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow><mi>P</mi><mi>u</mi><mi>t</mi><mi>t</mi><mi>i</mi><mi>n</mi><mi>g</mi> <mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi> <mi>o</mi><mi>f</mi> <mi>A</mi> <mi>a</mi><mi>n</mi><mi>d</mi> <mi>C</mi> <mi>i</mi><mi>n</mi> <mrow><mo>(</mo><mrow><mn>3</mn></mrow><mo>)</mo></mrow> <mi>w</mi><mi>e</mi> <mi>g</mi><mi>e</mi><mi>t</mi><mo>,</mo></mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>2</mn> <mo>×</mo> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>9</mn> </mrow> </mfrac> <mo>+</mo> <mn>2</mn> <mi>B</mi> <mo>−</mo> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>9</mn> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>⇒</mo> <mn>2</mn> <mi>B</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>9</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>4</mn> </mrow> <mrow> <mn>9</mn> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>⇒</mo> <mn>2</mn> <mi>B</mi> <mo>=</mo> <mfrac> <mrow> <mn>6</mn> </mrow> <mrow> <mn>9</mn> </mrow> </mfrac> <mo>:</mo> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>⇒</mo> <mi>B</mi> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </math> <div><div><picture><img src="https://images.shiksha.com/mediadata/images/articles/1742894750phpUWUP5l.jpeg" alt="" width="328" height="279"></picture></div></div>

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The value of $\int_{0}^{π / 4} \frac{x d x}{s i n^{4} (2 x) + c o s^{4} (2 x)}$ equal to

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