Mathematics

$$\displaystyle \int\frac{dx}{x(x^7+1)}$$ is equal to 


ANSWER

$$\dfrac{1}{7}log\left(\dfrac{x^7}{x^7+1}\right)+c$$


SOLUTION
$$\displaystyle I= \int\dfrac{dx}{x(x^7+1)}$$

multiplying and dividing by $$x^6$$ we get

$$\displaystyle I= \int\dfrac{x^6dx}{x^7(x^7+1)}$$

put $$x^7 = t\rightarrow7x^6dx =dt$$

$$\displaystyle I= \dfrac17\int\dfrac{dt}{t(t+1)}$$

$$\displaystyle I=\dfrac17 \int\dfrac{dt}{t^2+t}\Rightarrow\int\dfrac{1}{ \begin{pmatrix}t^2+t+\dfrac14-\dfrac14\end{pmatrix}}dt$$

$$I=\dfrac17\displaystyle  \int\dfrac{1}{(t+\dfrac12)^2-(\dfrac12)^2}dt$$

$$\displaystyle \Rightarrow I= \dfrac17\times\dfrac{1}{2\times\dfrac12}log\begin{pmatrix}\dfrac{t+\dfrac12-\dfrac12}{t+\dfrac12+\dfrac12}\end{pmatrix} + C$$

$$\begin{bmatrix}\because\displaystyle \int\dfrac{1}{x^2-a^2}dx =\dfrac{1}{2a}log\dfrac{x-a}{x+a}+ C\end{bmatrix}$$

$$\Rightarrow I= \dfrac17log\begin{pmatrix}\dfrac{t}{t+1}\end{pmatrix} + C$$

$$\Rightarrow I= \dfrac17log\begin{pmatrix}\dfrac{x^7}{x^7+1}\end{pmatrix} + C$$

$$\therefore \text {option B is correct} $$
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Single Correct Hard Published on 17th 09, 2020
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