Question

$\text{The integral}\int \frac{dx}{(x+1{)}^{\frac{3}{4}}(x-2{)}^{\frac{5}{4}}}\text{is equal to}$

• A. $4{\left(\frac{x+1}{x-2}\right)}^{\frac{1}{4}}+C$
• B. $-\frac{4}{3}{\left(\frac{x+1}{x-2}\right)}^{\frac{1}{4}}+C$
• C. $-\frac{4}{3}{\left(\frac{x-2}{x+1}\right)}^{\frac{1}{4}}+C$
• D. $4{\left(\frac{x-2}{x+1}\right)}^{\frac{1}{4}}+C$

Answer 1

The correct option is B $-\frac{4}{3}{\left(\frac{x+1}{x-2}\right)}^{\frac{1}{4}}+C$

$I=\int \frac{dx}{(x+1{)}^{\frac{3}{4}}(x-2{)}^{\frac{5}{4}}}=\int \frac{dx}{\left(\frac{x+1}{x-2}{)}^{\frac{3}{4}}\right(x-2{)}^2}\text{Put}\frac{x+1}{x-2}=z\Rightarrow \frac{-3}{(x-2{)}^2}dx=dzI=-\frac{1}{3}\int \frac{dz}{z^{\frac{3}{4}}}=-\frac{1}{3}\times \frac{z^{\frac{1}{4}}}{\frac{1}{4}}+C=-\frac{4}{3}(\frac{x+1}{x-2}{)}^{\frac{1}{4}}+C$