Question

If $\int x^{\frac{13}{2}}.{(1+x^{\frac{5}{2}})}^{\frac{1}{2}}dx=A{(1+x^{\frac{5}{2}})}^{\frac{7}{2}}+B{(1+x^{\frac{5}{2}})}^{\frac{5}{2}}+C{(1+x^{\frac{5}{2}})}^{\frac{3}{2}}$, then

• A. $A=-\frac{4}{35},B=-\frac{8}{25},C=\frac{4}{15}$
• B. $A=\frac{4}{35},B=-\frac{8}{25},C=-\frac{4}{15}$
• C. $A=\frac{4}{35},B=-\frac{8}{25},C=\frac{4}{15}$
• D. None of these

Answer 1

The correct option is C $A=\frac{4}{35},B=-\frac{8}{25},C=\frac{4}{15}$

Let $I=\int x^{\frac{13}{2}}.{(1+x^{\frac{5}{2}})}^{\frac{1}{2}}dx=\int x^5.x^{3/2}{(1+x^{\frac{5}{2}})}^{\frac{1}{2}}dx$
Substitute $1+x^{\frac{5}{2}}=z^2\Rightarrow \frac{5}{2}{x}^{3/2}dx=2zdz$
$\therefore I=\int x^5\frac{4}{5}.z.zdz=\frac{4}{5}\int z^2.{(z^2-1)}^{2}dz=\frac{4}{5}\int z^2(z^4-2z^2+1)dz$
$=\frac{4}{5}[\frac{z^7}{7}-\frac{2z^5}{5}+\frac{z^3}{3}]+c=\frac{4}{35}{(1+x^{\frac{5}{2}})}^{\frac{7}{2}}-\frac{8}{35}{(1+x^{\frac{5}{2}})}^{\frac{5}{2}}+\frac{4}{15}{(1+x^{\frac{5}{2}})}^{\frac{3}{2}}$
Hence $A=\frac{4}{35},B=-\frac{8}{25},C=\frac{4}{15}$