Question

If ∫ x^13/2 (1+ x^5/2)^1/2 dx = 2/5 [ 2/7 (1+x^5/2)^7/2 - A + 2/3 (1+ x^5/2)^3/2 ] + c

then A = ?

Answer 1

$$\begin{gathered} Ifinyourquestionthereis{x}^{\frac{3}{2}}insteadof{x}^{\frac{13}{2}},thenthesolutionisasfollows, \\ \int x^{\frac{3}{2}}{\left(1+x^{\frac{5}{2}}\right)}^{\frac{1}{2}}dx=\frac{2}{5}\int {\left(1+t\right)}^{\frac{1}{2}}dt \\ \left(let{x}^{\frac{5}{2}}=t\Rightarrow \frac{5}{2}{x}^{\frac{3}{2}}dx=dt\Rightarrow x^{\frac{3}{2}}dx=\frac{2}{5}dt\right) \\ =\frac{2}{5}\int {\left(1+t\right)}^{\frac{1}{2}}dt \\ =\frac{2}{5}\left(\frac{{\left(1+t\right)}^{\frac{3}{2}}}{{\displaystyle \frac{3}{2}}}\right) \\ =\frac{2}{5}\left(\frac{2}{3}{\left(1+x^{\frac{5}{2}}\right)}^{\frac{3}{2}}\right)+c \\ Now, \\ \frac{2}{5}\left(\frac{2}{3}{\left(1+x^{\frac{5}{2}}\right)}^{\frac{3}{2}}\right)+c=\frac{2}{5}\left(\frac{2}{7}{\left(1+x^{\frac{5}{2}}\right)}^{\frac{7}{2}}-A+\frac{2}{3}{\left(1+x^{\frac{5}{2}}\right)}^{\frac{3}{2}}\right)+c \\ Oncomparingwiththegivenexpression,\mathit{A}\mathbf{=}\frac{\mathbf{2}}{\mathbf{7}}{\left(\mathbf{1}\mathbf{+}\mathbf{}{\mathbf{x}}^{\frac{\mathbf{5}}{\mathbf{2}}}\right)}^{\frac{\mathbf{7}}{\mathbf{2}}} \end{gathered}$$