Question

Let $f\left(x\right)=\frac{x}{(1+x^7{)}^{\frac{1}{7}}}$ and $g\left(x\right)=\left(fofofofofofof\right)\left(x\right),\text{then}\int x^{5}g\left(x\right)dx$ is

• A. $\frac{1}{42}(1+6x^7{)}^{6/7}+c$
• B. $\frac{1}{35}(1+7x^7{)}^{5/7}+c$
• C. $\frac{1}{35}(1+5x^7{)}^{5/7}+c$
• D. $\frac{1}{42}(1+7x^7{)}^{6/7}+c$

Answer 1

The correct option is D $\frac{1}{42}(1+7x^7{)}^{6/7}+c$

$f\left(f\right(x\left)\right)=\frac{\frac{x}{(1+x^7{)}^{1/7}}}{(1+(\frac{x}{(1+x^7{)}^{1/7}}{)}^7{)}^{1/7}}=\frac{x}{(1+2x^7{)}^{1/7}}$
So $f_0{f}_0{f}_0{f}_0{f}_0{f}_{0}f\left(x\right)=\frac{x}{(1+7x^7{)}^{1/7}}$
So $I=\int \frac{x^6}{(1+7x^7{)}^{1/7}}dx$
$Let1+7x^7=t\Rightarrow 49x^{6}dx=dtI=\int \frac{dt}{49t^{1/7}}=\frac{1}{49}\frac{t^{6/7}}{6/7}+c=\frac{1}{42}(1+7x^7{)}^{6/7}+c$