Question

If $\int \frac{(\sqrt{x}{)}^5}{(\sqrt{x}{)}^7+x^6}dx=aln\left(\frac{x^k}{x^k+1}\right)+c$, the values of a and k respectively are

• A. $\frac{5}{2}$ and $\frac{2}{5}$
• B. $\frac{2}{5}$ and $\frac{5}{2}$
• C. $\frac{5}{2}$ and $2$
• D. None of these

Answer 1

The correct option is B $\frac{2}{5}$ and $\frac{5}{2}$

$I=\int \frac{dx}{\left(\sqrt{x}{)}^{+}\right(\sqrt{x}{)}^7}=\int \frac{dx}{(\sqrt{x}{)}^7[1+\frac{1}{(\sqrt{x}{)}^5}]}$
Put $\frac{1}{(\sqrt{x}{)}^5}=y\Rightarrow \frac{dy}{dx}=-\frac{5}{2\left(\sqrt{x}{)}^7\right)}\therefore I=\int \frac{-2dy}{5(1+y)}=-\frac{2}{5}ln|1+y|+c=\frac{2}{5}ln\left[\frac{(\sqrt{x}{)}^5}{(\sqrt{x}{)}^5+1}\right]+C\therefore a=\frac{2}{5},k=\frac{5}{2}$