Mathematics

f(x) = $$\displaystyle \int \frac{dx}{x^{2}\left ( 1+x^{5} \right )^{4/5}}$$, taking c = 0, value of $$|f(1)|=$$


ANSWER

2


SOLUTION
$$\displaystyle I=\int \frac{dx}{x^{2}\left ( 1+x^{5} \right )^{4/5}} \int \frac{dx}{x^{6}\left ( \frac{1}{x^{5}+1} \right )^{4/5}}$$
$$\displaystyle Let\:t=1+\frac{1}{x^{5}}or\:dt=\frac{5dx}{x^{6}}$$
$$\displaystyle \therefore I=-\frac{1}{5}\int \frac{dt}{t^{4/5}}=-t^{3/5}+c=-\left ( 1+\frac{1}{x^{5}} \right )^{3/5}+c$$
$$\displaystyle=-\left(\frac{1+x^{5}}{x}\right)^{\tfrac{3}{5}}+c$$

$$f(x)=-\left(\dfrac{1+x^{5}}{x}\right)^{\tfrac{3}{5}}\Rightarrow f(1)=-2\Rightarrow |f(1)|=2$$
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One Word Hard Published on 17th 09, 2020
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