∫1(2x+1)56(3x+5)76dx=

∫1(2x+1)56(3x+5)76dx =∫12x+13x+5563x+556+76dx =∫12x+13x+55/63x+52dx =17∫dtt5/6 put 2x+13x+5-t =17t1/61/6+C 3x+52-2x+133x+52dx=dt =672x+13x+51/6+C 73x+52dx=dt

∫1(2x+1)56(3x+5)76dx=

∫1(2x+1)56(3x+5)76dx =∫12x+13x+5563x+556+76dx =∫12x+13x+55/63x+52dx =17∫dtt5/6 put 2x+13x+5-t =17t1/61/6+C 3x+52-2x+133x+52dx=dt =672x+13x+51/6+C 73x+52dx=dt

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$\int \frac{1}{{(2 x + 1)}^{\frac{5}{6}} {(3 x + 5)}^{\frac{7}{6}}} d x =$

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$\frac{6}{7} {(\frac{2 x + 1}{3 x + 5})}^{\frac{1}{6}} + c$

$\frac{6}{7} {(\frac{3 x + 5}{2 x + 1})}^{\frac{1}{6}} + c$

$- \frac{6}{7} {(\frac{3 x + 5}{2 x + 1})}^{\frac{1}{6}} + c$

$- \frac{6}{7} {(\frac{2 x + 1}{3 x + 5})}^{\frac{1}{6}} + c$

answer is A.

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Detailed Solution

$\int \frac{1}{{(2 x + 1)}^{\frac{5}{6}} {(3 x + 5)}^{\frac{7}{6}}} d x = \int \frac{1}{{(\frac{2 x + 1}{3 x + 5})}^{\frac{5}{6}} {(3 x + 5)}^{\frac{5}{6} + \frac{7}{6}}} d x = \int \frac{1}{{(\frac{2 x + 1}{3 x + 5})}^{5 / 6} {(3 x + 5)}^{2}} d x = \frac{1}{7} \int \frac{d t}{t^{5 / 6}} p u t \frac{2 x + 1}{3 x + 5} - t = \frac{1}{7} \frac{t^{1 / 6}}{1 / 6} + C \frac{(3 x + 5) 2 - (2 x + 1) 3}{{(3 x + 5)}^{2}} d x = d t = \frac{6}{7} {(\frac{2 x + 1}{3 x + 5})}^{1 / 6} + C \frac{7}{{(3 x + 5)}^{2}} d x = d t$