Question

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

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

Answer 1

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

$I=\int \frac{x^4(5+4x)}{(x^5+x+1{)}^2}dx\Rightarrow I=\int \frac{x^4(5+4x)}{x^{10}(1+x^{-4}+x^{-5}{)}^2}dx\Rightarrow I=\int \frac{\frac{(5x^4+4x^5)}{x^{10}}}{(1+x^{-4}+x^{-5}{)}^2}dx$
$\Rightarrow I=\int \frac{5x^{-6}+4x^{-5}}{(1+x^{-4}+x^{-5}{)}^2}dx$
Put $(1+x^{-4}+x^{-5})=t\Rightarrow -(5x^{-6}+4x^{-5})dx=dt$

$\Rightarrow I=-\int \frac{dt}{t^2}=\frac{1}{t}+c$
$\Rightarrow I=\frac{1}{1+x^{-4}+x^{-5}}+c$