Question

If $\int f\left(x\right)dx=\frac{{\left(\log x\right)}^5}{5}+c$, then $f\left(x\right)=$_______

• A. $\frac{{\left(\log x\right)}^6}{6}$
• B. $\frac{\log x^4}{5}$
• C. $\frac{{\left(\log x\right)}^4}{x}$
• D. $\frac{{\left(\log x\right)}^{}}{4}$

Answer 1

The correct option is C $\frac{{\left(\log x\right)}^4}{x}$

As differentiation is inverse of integration so f(x) can be found by differentiating $\frac{(logx{)}^5}{5}$

$\frac{d}{dx}\frac{(logx{)}^5}{5}=\frac{1}{5}\frac{d}{dx}(logx{)}^5$

Differentiating by chain rule

$\frac{1}{5}\frac{d}{dx}(logx{)}^5=\frac{1}{5}\times 5(logx{)}^4\frac{d}{dx}logx=(logx{)}^4\frac{1}{x}=\frac{(logx{)}^4}{x}$

Hence the correct answer is $\frac{(logx{)}^4}{x}$