Question

Hey Guys ...Plss Solve this Integration ONLY BEST answer is required ✌✌​

Answer 1

Answer:

$\large\boxed{\sf{ -\dfrac{1}{ \sqrt{ {x}^{2} + {a}^{2} } } + c}}$

Explanation:

To integrate the given expression,

$\displaystyle\int \dfrac{xdx}{ {( {x}^{2} + {a}^{2}) }^{ \frac{3}{2} } }$

Let's assume that,

${x}^{2} + {a}^{2} = {t}^{2}$

On differentiatiating both sides, we get,

$= > 2xdx = 2tdt \\ \\ = > xdx = tdt$

Substituting the values in given integral, we get,

$= \displaystyle\int \dfrac{tdt}{ {( {t}^{2}) }^{ \frac{3}{2} } } \\ \\ = \displaystyle\int \dfrac{t}{ {t}^{3} } dt \\ \\ = \displaystyle\int {t}^{ - 2} dt$

But, we know that,

• $\displaystyle\int {x}^{n} = \dfrac{ {x}^{n + 1} }{n + 1}$

Therefore, we will get,

$= \dfrac{ {t}^{ - 2 + 1} }{ - 2 + 1} + c \\ \\ = \dfrac{ {t}^{ - 1} }{ - 1} + c \\ \\ = - \dfrac{1}{t} + c$

Substituting the value of t, we get,

$= - \dfrac{1}{ \sqrt{ {x}^{2} + {a}^{2} } } + c$

Where, c is any arbitrary constant.