Question

Integrate wrt x $\frac{1}{\sqrt{(3x+4)}-\sqrt{(3x+1)}}+\frac{8x+13}{\sqrt{(4x+7)}}+(7x-2)\sqrt{3x+2}.$

• A. $\frac{1}{3}[\frac{1}{3}.\frac{2}{3}{(3x+4)}^{3/2}+\frac{1}{3}.\frac{2}{3}{(3x+1)}^{3/2}+2.\frac{1}{4}.\frac{2}{3}{(4x+7)}^{3/2}-\frac{1}{4}.2\sqrt{4x+7}]$
  $+\frac{7}{3}[\frac{1}{3}.\frac{2}{5}{(3x+2)}^{5/2}-\frac{20}{7}.\frac{1}{3}.\frac{2}{3}{(3x+2)}^{3/2}]$
• B. $\frac{1}{3}[\frac{1}{3}.\frac{2}{3}{(3x+4)}^{3/2}+\frac{1}{3}.\frac{2}{3}{(3x+1)}^{3/2}+2.\frac{1}{4}.\frac{2}{3}{(4x+7)}^{3/2}-\frac{1}{4}.2\sqrt{4x+7}]$
  $-\frac{7}{9}[\frac{1}{3}.\frac{2}{5}{(3x+2)}^{5/2}-\frac{20}{7}.\frac{1}{3}.\frac{2}{3}{(3x+2)}^{3/2}]$
• C. $\frac{1}{3}[\frac{1}{3}.\frac{2}{3}{(3x+4)}^{3/2}-\frac{1}{3}.\frac{2}{5}{(3x+1)}^{3/2}+2.\frac{1}{4}.\frac{2}{3}{(4x+7)}^{3/2}-\frac{1}{4}.2\sqrt{4x+7}]$
  $-\frac{5}{3}[\frac{1}{3}.\frac{2}{5}{(3x+2)}^{5/2}-\frac{20}{7}.\frac{1}{3}.\frac{2}{3}{(3x+2)}^{3/2}]$
• D. None

Answer 1

The correct option is A $\frac{1}{3}[\frac{1}{3}.\frac{2}{3}{(3x+4)}^{3/2}+\frac{1}{3}.\frac{2}{3}{(3x+1)}^{3/2}+2.\frac{1}{4}.\frac{2}{3}{(4x+7)}^{3/2}-\frac{1}{4}.2\sqrt{4x+7}]$

$+\frac{7}{3}[\frac{1}{3}.\frac{2}{5}{(3x+2)}^{5/2}-\frac{20}{7}.\frac{1}{3}.\frac{2}{3}{(3x+2)}^{3/2}]$
(i)$\frac{1}{\sqrt{(3x+4)-\sqrt{(3x+1)}}}=\frac{\sqrt{3x+4}+\sqrt{3x+1}}{(3x+4)-(3x+1)}.$
$=\frac{1}{3}[\sqrt{3x+4}+\sqrt{3x+1}].$
(ii)$\frac{8x+13}{\sqrt{(4x+7)}}=\frac{2(4x+7)-1}{\sqrt{(4x+7)}}=2\sqrt{4x+7}-\frac{1}{\sqrt{4x+7}}$
(iii)$(7x-2)\sqrt{(3x+2)}=\frac{7}{3}(3x-\frac{6}{7})\sqrt{3x+2}$
Hence $I=\frac{7}{3}(3x+2-\frac{6}{7}-2)\sqrt{3x+2}.$
$=\frac{7}{3}[{(3x+2)}^{3/2}-\frac{20}{7}\sqrt{3x+2}].$
$\frac{1}{3}[\frac{1}{3}.\frac{2}{3}{(3x+4)}^{3/2}+\frac{1}{3}.\frac{2}{3}{(3x+1)}^{3/2}+2.\frac{1}{4}.\frac{2}{3}{(4x+7)}^{3/2}-\frac{1}{4}.2\sqrt{4x+7}]$
$+\frac{7}{3}[\frac{1}{3}.\frac{2}{5}{(3x+2)}^{5/2}-\frac{20}{7}.\frac{1}{3}.\frac{2}{3}{(3x+2)}^{3/2}].$