Question

The derivative of $x^{-4}+x^{-5}$ is $-(4x^{-5}+5x^{-6})$. So, $\int \frac{5x^4+4x^5}{(x^5+x+1{)}^2}dx$ is equal to

• A. $x^5+x+1+c$
• B. $\frac{1}{x^5+x+1}+c$
• C. $x^{-4}+x^{-5}+c$
• D. $\frac{x^5}{x^5+x+1}+c$

Answer 1

The correct option is D $\frac{x^5}{x^5+x+1}+c$

$\int \frac{5x^4+4x^5}{(x^5+x+1{)}^2}dx$

$=\int \frac{5x^4+4x^5}{x^{10}(1+x^{-4}+x^{-5}{)}^2}dx$

$=-\int \frac{-(5x^{-6}+4x^{-5})}{(1+x^{-4}+x^{-5}{)}^2}dx$

$=-\int \frac{d(x^{-5}+x^{-4}+1)}{(1+x^{-4}+x^{-5}{)}^2}dx$

$=\frac{1}{(1+x^{-4}+x^{-5})}+c$ [ Where $c$ being integrating constant]

$=\frac{x^5}{(1+x+x^5)}+c$