Question

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

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

Answer 1

The correct option is A $I=\frac{x^5}{x^5+x+1}+C$

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