Question

If $\int x^{13/2}.{(1+x^{5/2})}^{1/2}dx=p{(1+x^{5/2})}^{7/2}+Q{(1+x^{5/2})}^{5/2}+R{(1+x^{5/2})}^{3/2}+C$, then P, Q and R are

• A. $P=\frac{4}{35},Q=-\frac{8}{25},R=\frac{4}{15}$
• B. $P=\frac{4}{35},Q=\frac{8}{25},R=\frac{4}{15}$
• C. $P=-\frac{4}{35},Q=-\frac{8}{25},R=\frac{4}{15}$
• D. $P=\frac{4}{35},Q=-\frac{8}{25},R=-\frac{4}{15}$

Answer 1

The correct option is A $P=\frac{4}{35},Q=-\frac{8}{25},R=\frac{4}{15}$

$I=\int x^{\frac{13}{2}}{(1+x^{\frac{5}{2}})}^{\frac{1}{2}}dx$

Let $t=1+x^{\frac{5}{2}}\Rightarrow dt=\frac{5}{2}{x}^{\frac{5}{2}-1}dx=\frac{5}{2}{x}^{\frac{3}{2}}dx$

$\Rightarrow \frac{2}{5}\frac{dt}{x^{\frac{3}{2}}}=dx$

We have $t=1+x^{\frac{5}{2}}$

$\Rightarrow t-1=x^{\frac{5}{2}}$

$\Rightarrow x^5={(t-1)}^2$

$I=\int x^{\frac{13}{2}}{(1+x^{\frac{5}{2}})}^{\frac{1}{2}}dx$

$=\int x^{\frac{13}{2}}{t}^{\frac{1}{2}}\frac{2}{5}\frac{dt}{x^{\frac{3}{2}}}$

$=\frac{2}{5}\int x^{\frac{13-3}{2}}{t}^{\frac{1}{2}}dt$

$=\frac{2}{5}\int x^{\frac{10}{2}}{t}^{\frac{1}{2}}dt$

$=\frac{2}{5}\int x^5{t}^{\frac{1}{2}}dt$

$=\frac{2}{5}\int {(t-1)}^2{t}^{\frac{1}{2}}dt$ where ${(t-1)}^2=x^5$

$=\frac{2}{5}\int (t^2-2t+1)t^{\frac{1}{2}}dt$

$=\frac{2}{5}\int t^{2+\frac{1}{2}}dt-\frac{4}{5}\int t^{1+\frac{1}{2}}dt+\frac{2}{5}\int t^{\frac{1}{2}}dt$

$=\frac{2}{5}\int t^{\frac{5}{2}}dt-\frac{4}{5}\int t^{\frac{3}{2}}dt+\frac{2}{5}\int t^{\frac{1}{2}}dt$

$=\frac{2}{5}\frac{t^{\frac{5}{2}+1}}{\frac{5}{2}+1}-\frac{4}{5}\frac{t^{\frac{3}{2}+1}}{\frac{3}{2}+1}+\frac{2}{5}\frac{t^{\frac{1}{2}+1}}{\frac{1}{2}+1}+c$

$=\frac{2}{5}\frac{t^{\frac{7}{2}}}{\frac{7}{2}}-\frac{4}{5}\frac{t^{\frac{5}{2}}}{\frac{5}{2}}+\frac{2}{5}\frac{t^{\frac{3}{2}}}{\frac{3}{2}}+c$

$=\frac{2}{5}\frac{2t^{\frac{7}{2}}}{7}-\frac{4}{5}\frac{2t^{\frac{5}{2}}}{5}+\frac{2}{5}\frac{2t^{\frac{3}{2}}}{3}+c$

$=\frac{4}{35}{t}^{\frac{7}{2}}-\frac{8}{25}{t}^{\frac{5}{2}}+\frac{4}{15}{t}^{\frac{3}{2}}+c$

where $t=1+x^{\frac{5}{2}}$

$=\frac{4}{35}{(1+x^{\frac{5}{2}})}^{\frac{7}{2}}-\frac{8}{25}{(1+x^{\frac{5}{2}})}^{\frac{5}{2}}+\frac{4}{15}{(1+x^{\frac{5}{2}})}^{\frac{3}{2}}+c$

is of the form $P{(1+x^{\frac{5}{2}})}^{\frac{7}{2}}+Q{(1+x^{\frac{5}{2}})}^{\frac{5}{2}}+R{(1+x^{\frac{5}{2}})}^{\frac{3}{2}}+c$

$\therefore P=\frac{4}{35},Q=-\frac{8}{25}$ and $R=\frac{4}{15}$