Question

$I=\int \frac{(\sqrt{x}{)}^3}{(\sqrt{x}{)}^5+x^4}dx=A\ln \left|\frac{x^K}{x^K+1}\right|+C$
(where $A,k$ are fixed constants and $C$ is integration constant)

• A. $3A+2K=5$
• B. $3A+2K=6$
• C. $3K-2A=1$
• D. $2K-3A=1$

Answer 1

Answer: (A), (D)

$2K-3A=1$

$I=\int \frac{(\sqrt{x}{)}^3}{(\sqrt{x}{)}^5+x^4}dx=\int \frac{dx}{(\sqrt{x}{)}^2+(\sqrt{x}{)}^5}=\int \frac{dx}{x(x^{\frac{3}{2}}+1)}=\int \frac{dx}{x^{\frac{5}{2}}(1+x^{\frac{-3}{2}})}$
Put $1+x^{\frac{-3}{2}}=t$
$\Rightarrow -\frac{3}{2}{x}^{\frac{-5}{2}}dx=dt$
$\Rightarrow I=\int -\frac{2}{3}\frac{dt}{t}=-\frac{2}{3}\ln |t|+C=-\frac{2}{3}\ln |1+x^{\frac{-3}{2}}|+C=\frac{2}{3}\ln \left|\frac{x^{\frac{3}{2}}}{x^{\frac{3}{2}}+1}\right|+C$
$\Rightarrow A=\frac{2}{3},K=\frac{3}{2}$