Question

Evaluate: $\int \frac{5x^8+7x^6}{{(x^2+1+2x^7)}^2}dx$

• A. $\frac{2x^7}{2x^7+x^2+1}$
• B. $\frac{x^7}{2x^7+x^2+1}$
• C. $\frac{x^6}{2x^7+x^2+1}$
• D. $\frac{x^{14}}{2x^7+x^2+1}$

Answer 1

The correct option is B $\frac{x^7}{2x^7+x^2+1}$

Let $I=\int \frac{5x^8+7x^6}{{(x^2+1+2x^7)}^2}dx$
$=\int \frac{5x^8+7x^6}{x^{14}{[\frac{1}{x^5}+\frac{1}{x^7}+2]}^2}dx=\int \frac{[\frac{5}{x^6}+\frac{7}{x^8}]}{{[\frac{1}{x^5+\frac{1}{x^7}}+2]}^2}dx$
Put $\frac{1}{x^5}+\frac{1}{x^7}+2=t\Rightarrow -[\frac{5}{x^6}+\frac{7}{x^8}]dx=dt$
Therefore
$I=\int -\frac{1}{t^2}dt=\frac{1}{t}={[\frac{1}{x^5}+\frac{1}{x^7}+2]}^{-1}$
$={\left[\frac{2x^7+x^2+1}{x^7}\right]}^{-1}=\frac{x^7}{2x^7+x^2+1}$