Question

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

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

Answer 1

Answer: (C)

$-\frac{4}{3}{\left(\frac{x+1}{x-2}\right)}^{1/4}+C$

Given

$\int \frac{1}{(x-5{)}^{\frac{5}{4}}(x+1{)}^{\frac{3}{4}}}dx$

Substitute $u=\sqrt[4]{4x-2}$ $dx=4(x-2{)}^{\frac{3}{4}}du$

$=4\int \frac{1}{u^2(u^4+3{)}^{\frac{3}{4}}}du$

Substitute $v=\frac{\sqrt[4]{4u^4+3}}{u}$ , $du=\frac{1}{\frac{u^2}{(u^4+3{)}^{\frac{3}{4}}}-\frac{\sqrt[4]{4u^4+3}}{u^2}}dv$

$=\int -\frac{1}{3}dv$

$=-\frac{1}{3}\int dv$

$=-\frac{v}{3}$

$=-\frac{\sqrt[4]{4u^4+3}}{3u}$

$=-\frac{4\sqrt[4]{4u^4+3}}{3u}$

$=-\frac{4\sqrt[4]{4x+1}}{3\sqrt[4]{4x-2}}$

$=-\frac{4}{3}{\left(\frac{x+1}{x-2}\right)}^{\frac{1}{4}}$