Question

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

• A. $\frac{-1}{7}\ln (1+x^{-7})+C$
• B. $\frac{1}{7}\ln (1+x^{-7})+C$
• C. $\frac{-1}{7}\ln (1+x^7)+C$
• D. $\frac{1}{7}\ln (1+x^7)+C$

Answer 1

The correct option is D $\frac{1}{7}\ln (1+x^7)+C$

We are given thta $I=\int \frac{x^6}{1+x^7}dx$

Let

$1+x^7=t$

now, differentiating above equation wrt $x$ we get

$7x^{6}dx=dt$

$\therefore x^{6}dx=\frac{dt}{7}$

$I=\int \frac{dt}{7\left(t\right)}=\frac{7}{7}\int \frac{dt}{t}$

$I=\frac{1}{7}\ln |t|+C$

here $t=1+x^7$

$I=\frac{1}{7}\ln |1+x^7|+C$