Mathematics

Integrate:$$\displaystyle \dfrac {x\cdot \sec^{2}(x^{2})}{\sqrt {\tan^{3}(x^{2})}}$$


SOLUTION

Consider the given integral.

$$I=\displaystyle \int{\dfrac{x{{\sec }^{2}}\left( {{x}^{2}} \right)}{{{\tan }^{3}}\left( {{x}^{2}} \right)}dx}$$

 

Let $$t=\tan \left( {{x}^{2}} \right)$$

$$ \dfrac{dt}{dx}={{\sec }^{2}}\left( {{x}^{2}} \right)\times 2x $$

$$ \dfrac{dt}{2}=x{{\sec }^{2}}\left( {{x}^{2}} \right)dx $$

 

Therefore,

$$ I=\dfrac{1}{2}\int{\dfrac{1}{{{t}^{3}}}dt} $$

$$ I=\dfrac{1}{2}\left( -\dfrac{1}{2{{t}^{2}}} \right)+C $$

$$ I=-\dfrac{1}{4{{t}^{2}}}+C $$

 

On putting the value of $$t$$, we get

$$ I=-\dfrac{1}{4{{\left( \tan \left( {{x}^{2}} \right) \right)}^{2}}}+C $$

$$ I=-\dfrac{1}{4{{\tan }^{2}}\left( {{x}^{2}} \right)}+C $$

 

Hence, this is the answer.

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