Question

The value of $\int \frac{(x-2)dx}{{\left\{\right(x-2{)}^2(x+3{)}^7\}}^{1/3}}$ is

• A. $\frac{3}{20}{\left(\frac{x-2}{x+3}\right)}^{4/3}+c$
• B. $\frac{3}{20}{\left(\frac{x-2}{x+3}\right)}^{3/4}+c$
• C. $\frac{5}{12}{\left(\frac{x-2}{x+3}\right)}^{4/3}+c$
• D. $\frac{3}{20}{\left(\frac{x-2}{x+3}\right)}^{5/3}+c$

Answer 1

The correct option is A $\frac{3}{20}{\left(\frac{x-2}{x+3}\right)}^{4/3}+c$

$I=\int \frac{(x-2)dx}{{\left\{\right(x-2{)}^2(x+3{)}^7\}}^{1/3}}$
$\Rightarrow I=\int \frac{dx}{(x-2{)}^{-1/3}(x+3{)}^{7/3}}$
$\Rightarrow I=\int \frac{dx}{(x-2{)}^2{\left(\frac{x+3}{x-2}\right)}^{7/3}}$
Let $\frac{x+3}{x-2}=t$
Differentiating both sides, we get
$\frac{-5dx}{(x-2{)}^2}=dt$
$\therefore I=-\frac{1}{5}\int \frac{dt}{t^{7/3}}$
$=\frac{3}{20t^{4/3}}+c$

$=\frac{3}{20}{\left(\frac{x-2}{x+3}\right)}^{4/3}+c$