Mathematics

The evaluation of $$\displaystyle\int{\frac{px^{\displaystyle p+2q-1}-qx^{\displaystyle q-1}}{x^{\displaystyle 2p+2q}+2x^{\displaystyle p+q}+1}dx}$$ is


ANSWER

$$\displaystyle-\frac{x^q}{x^{\displaystyle p+q}+1}+C$$


SOLUTION
$$\displaystyle I=\int { \frac { { px }^{ p+2q-1 }-q{ x }^{ q-1 } }{ { x }^{ 2p+2q }+{ 2x }^{ p+q }+1 } dx } =\int { \frac { { px }^{ p-1 }-q{ x }^{ -q-1 } }{ { \left( { x }^{ p }+{ x }^{ -q } \right)  }^{ 2 } } dx } .$$

Substitute $${ x }^{ q }+{ x }^{ -q }=t\Rightarrow \left( { px }^{ p-1 }-{ qx }^{ -q-1 } \right) dx=dt$$

$$\displaystyle I=\int { \frac { 1 }{ t }  } =-\frac { 1 }{ { x }^{ p }+{ x }^{ -q } } =-\frac { { x }^{ q } }{ { x }^{ p+q }+1. } $$
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Single Correct Hard Published on 17th 09, 2020
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