Question #302c7

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Answer 1 · 2016-01-03T21:18:17

$10e^sqrtx -3/2ln^2(cos(x/3))+C$

For simplicity, let:

$I_1=int ((5e^sqrtx)/sqrtx)dx$

and $I_2=int(ln(cos(x/3))/cot(x/3))dx$

$I=I_1+I_2$

$I_1=int((5e^sqrtx)/sqrtx)dx$

Let $t=sqrtx$
then $dt=1/(2sqrtx)dx$

$dx=(2t) dt$

$I_1=int(5e^t/cancelt*2cancelt) dt$

$I_1=10int(e^t)$

$I_1=10e^t + c_1=10e^sqrtx + c_1$

$I_2=int(ln(cos(x/3))/cot(x/3))dx$

Let $u=cos(x/3)$

then $du=-sin(x/3)/3dx$

$dx=-3/sin(x/3)du$

$I_2=-3int(lnu/u)du$

$I_2=-3ln^2u/2 + c_2=-3ln^2cos(x/3)/2 + c_2$

$I=I_1+I_2$

$I=10e^sqrtx -3ln^2cos(x/3)/2 + (c_1+c_2)$

$I=10e^sqrtx -3ln^2cos(x/3)/2 + C$ where $C=c_1+c_2$