Question

The value of $\int \frac{1-x^7}{x(1+x^7)}dx$ is equal to

• A. $\ln \left|x\right|+\frac{2}{7}\ln |1+x^7|+C$
• B. $\ln \left|x\right|-\frac{2}{7}\ln |1-x^7|+C$
• C. $\ln \left|x\right|-\frac{2}{7}\ln |1+x^7|+C$
• D. $\ln \left|x\right|+\frac{2}{7}\ln |1-x^7|+C$

Answer 1

The correct option is C $\ln \left|x\right|-\frac{2}{7}\ln |1+x^7|+C$

$\frac{1-x^7}{x(1+x^7)}=\frac{(1+x^7)-2x^7}{x(1+x^7)}=\frac{1}{x}-\frac{2x^6}{1+x^7}$

$\int \frac{1-x^7}{x(1+x^7)}dx=\int \frac{1}{x}dx-2\int \frac{x^6}{1+x^7}dx$

$=\int \frac{1}{x}dx-\frac{2}{7}\int \frac{7x^6}{1+x^7}dx$

$=\ln \mid x\mid -\frac{2}{7}\ln \mid 1+x^7\mid +C$ ........ $\int \frac{f^{\prime }\left(x\right)}{f\left(x\right)}dx=\log \left(f\right(x\left)\right)+c$