Question

Evaluate $\int \frac{3x^2+7}{x^8+3x^3+5x}dx$
(where $C$ is constant of integration)

• A. $-\frac{2}{5}\ln |1+\frac{3}{x^5}+\frac{5}{x^7}|+C$
• B. $\frac{1}{5}\ln |1+\frac{3}{x^5}+\frac{5}{x^7}|+C$
• C. $\frac{2}{5}\ln |1+\frac{3}{x^5}+\frac{5}{x^7}|+C$
• D. $-\frac{1}{5}\ln |1+\frac{3}{x^5}+\frac{5}{x^7}|+C$

Answer 1

The correct option is D $-\frac{1}{5}\ln |1+\frac{3}{x^5}+\frac{5}{x^7}|+C$

Let $I=\int \frac{3x^2+7}{x^8+3x^3+5x}dx$
$\Rightarrow I=\int \frac{3x^2+7dx}{x^8(1+3x^{-5}+5x^{-7})}$

$\Rightarrow I=\int \frac{3x^{-6}+7x^{-8}}{1+3x^{-5}+5x^{-7}}dx$
Put : $1+3x^{-5}+5x^{-7}=t$
$\Rightarrow (-15x^{-6}-35x^{-8})dx=dt$
$\Rightarrow -5(3x^{-6}+7x^{-8})dx=dt$

$\Rightarrow I=\int \frac{dt}{-5t}=-\frac{1}{5}\ln |t|+C$
$\Rightarrow I=-\frac{1}{5}\ln |1+\frac{3}{x^5}+\frac{5}{x^7}|+C$