Mathematics

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


ANSWER

$$P=\cfrac { 4 }{ 35 } ,Q=-\cfrac { 8 }{ 25 } ,R=\cfrac { 4 }{ 15 } $$


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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