Mathematics

$$\int {\dfrac{1}{{9{x^2} - 25}}dx = \_\_\_\_\_\_ + c.} $$


ANSWER

$$\dfrac{1}{{30}}\log \left| {\dfrac{{3x - 5}}{{3x + 5}}} \right|$$


SOLUTION
$$I=\int { \dfrac { 1 }{ {9{ x^{ 2 } }-25 } } dx } $$

$$ I=\dfrac { 1 }{ 9 } \int { \dfrac { 1 }{ { \left( { { x^{ 2 } }-\dfrac { { 25 } }{ 9 }  } \right)  } } dx } $$
$$ I=\dfrac { 1 }{ 9 } \int { \dfrac { 1 }{ {  { { x^{ 2 } }-\left(\dfrac { 5  }{ 3 }\right)^2  }  } } dx } $$

We know that
$$\int \dfrac { dx }{  x^ 2-a^2}=\dfrac{1}{2a}\log\left(\dfrac{x-a}{x+a}\right)+C $$

Therefore,
$$I =\dfrac { 1 }{ 9 } .\dfrac { 1 }{ { 2\times \dfrac { 5 }{ 3 }  } } \log  \left| { \dfrac { { x-\dfrac { 5 }{ 3 }  } }{ { x+\dfrac { 5 }{ 3 }  } }  } \right| +C$$
$$ I=\dfrac { 3 }{ { 90 } } \log  \left| { \dfrac { { \dfrac { { 3x-5 } }{ 3 }  } }{ { \dfrac { { 3x+5 } }{ 3 }  } }  } \right| +C$$
$$ I =\dfrac { 1 }{ { 30 } } \log  \left| { \dfrac { { 3x-5 } }{ { 3x+5 } }  } \right| +C$$

Hence, this is the answer.
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Single Correct Medium Published on 17th 09, 2020
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