Mathematics

Solve:$$\int\displaystyle\dfrac{x}{x^4-a^4}$$


SOLUTION
$$\int { \dfrac { x }{ { x }^{ 4 }-{ a }^{ 4 } } dx } \\ Let\quad { x }^{ 2 }=t\\ differentiating\quad w.r.t\quad x,\quad we\quad get\\ 2xdx=dt\quad \\ Now,\\ \int { \dfrac { x }{ { x }^{ 4 }-{ a }^{ 4 } } dx } \\ =\int { \dfrac { \dfrac { 1 }{ 2 } dt }{ { t }^{ 2 }-({ a }^{ 2 })^{ 2 } }  } \\ =\dfrac { 1 }{ 2 } \int { \dfrac { dt }{ { t }^{ 2 }-({ a }^{ 2 })^{ 2 } }  } \\ =\dfrac { 1 }{ 2 } \left[ \dfrac { 1 }{ { 2a }^{ 2 } } \log\left| \dfrac { t-{ a }^{ 2 } }{ t-{ a }^{ 2 } }  \right|  \right] +C\\ =\dfrac { 1 }{ { 4a }^{ 2 } } \log\left| \dfrac { x-{ a }^{ 2 } }{ x-{ a }^{ 2 } }  \right| +C$$
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Subjective Medium Published on 17th 09, 2020
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