Mathematics

Solve:$\int{\displaystyle \dfrac {dx}{x(x^{n}-t)}}$

SOLUTION
$\int{\cfrac{dx}{x \left( {x}^{n} - t \right)}}$
$= \int{\cfrac{dx}{{x}^{n+1} \left( 1 - \cfrac{t}{{x}^{n}} \right)}}$
Now,
Let $1 - \cfrac{t}{{x}^{n}} = u \\ \Rightarrow 0 + \cfrac{nt}{{x}^{n + 1}} dx = du \\ \Rightarrow \cfrac{dx}{{x}^{n+1}} = \cfrac{du}{nt}$
Therefore,
$\int{\cfrac{du}{nt}} = \cfrac{1}{nt} \int{du} = \cfrac{u}{nt}$
$\because u = 1 - \cfrac{t}{{x}^{n}}$
$\therefore \int{\cfrac{dx}{x \left( {x}^{n} - t \right)}} = \cfrac{1 - \cfrac{t}{{x}^{n}}}{nt} = \cfrac{{x}^{n} - t}{nt \cdot {x}^{n}}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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