Question

$\int \frac{x^3}{25x^8-16}dx.$

• A. $\frac{1}{160}\log \frac{5x^4+4}{5x^4-4}$
• B. $\frac{1}{80}\log \frac{5x^4-4}{5x^4+4}$
• C. $\frac{1}{160}\log \frac{5x^4-4}{5x^4+4}$
• D. $\frac{1}{40}\log \frac{5x^4+4}{5x^4-4}$

Answer 1

The correct option is C $\frac{1}{160}\log \frac{5x^4-4}{5x^4+4}$

Let $I=\int \frac{x^3}{25x^8-16}dx$
Put $5x^4=t\Rightarrow 20x^{3}dx=dt$
$I=\frac{1}{20}\int \frac{dt}{t^2-4^2}=\frac{1}{20}.\frac{1}{2.4}\log \frac{t-4}{t+4}$
$=\frac{1}{160}\log \frac{5x^4-4}{5x^4+4}$
Hence, option 'C' is correct.