Question

Question #302c7

Answer 1

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

Explanation:

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`

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`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`

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`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`

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`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`