Mathematics

$$I = \int \dfrac {1}{x(x^{6}+1)}dx$$


SOLUTION
$$I = \int {\dfrac{{\rm{1}}}{{x\left( {{x^{\rm{6}}} + {\rm{1}}} \right)}}{\rm{dx}}} $$
putting $$x^6+1=t$$
$$6x^5dx=dt$$
$$dx=\dfrac{dt}{6x^5}$$
$$I = \int {\dfrac{{dt}}{{6{x^5} \times x \times t}}} $$
$$I = \dfrac{1}{6}\int {\frac{{dt}}{{t\left( {t - 1} \right)}}} $$
$$I = \dfrac{1}{6}\int {\left( {\dfrac{1}{{t - 1}} - \dfrac{1}{t}} \right)dt} $$
$$I = \dfrac{1}{6}\int {\frac{1}{{t - 1}}dt}  - \dfrac{1}{6}\int {\dfrac{1}{t}dt} $$
$$I =\dfrac{1}{6}\log (t-1)-\dfrac{1}{6}\log (t)+C$$
$$I =\dfrac{1}{6}\log (\dfrac{t-1}{t})+C$$
$$I =\dfrac{1}{6}\log (\dfrac{x^6}{x^6+1})+C$$
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Subjective Medium Published on 17th 09, 2020
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