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
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