can anyone solve these integrals?

Question

Calculate the following integrals.

If the answer requires a constant of integration, enter it as c.

You may enter a symbolic answer for any of the following (for some, that means a formula in terms of x, for others, that means a formula for the exact answer).

For those integrals that have a numerical answer, you may enter the numerical value of the integral as a decimal, but be sure to enter sufficiently many decimal places (enter as many as you can).

#1 (S is supposed to represent the integral)
5
S (3/x)+(2 sqrt(x))dx
3

#2
5
S (5 e^(-x/2) - 4) dx
3


THANK YOU!

Answer 1

1) int(3x^-1 + 2x^1/2)dx

integrate this to give you 3lnx + 2*2*x^(3/2) / 3
replace x values with 5 and 3

(3ln(5) + 4(5)^(3/2) / 3) - (3ln(3) + 4(3)^(3/2) / 3)

calculate this to give you your answer

second one

int(5*e^(-x/2) -4)dx

lets say u = -x/2
than du = -dx/2, so dx = -2du

than you have

-2*int(5*e^(u))du - int(4)dx

-2*5*e^(u) - 4x
-10e^(-x/2) - 4x
replace x values of 5 and 3 in this integral

(-10*e^(-5/3) - 4(5)) - (-10*e^(-3/3) - 4(3))

calculate this for your answer

Answer 2

Doing integrals with typing sucks! But anyway.

S(3/x) + 2sqrt(x) dx would be...
3 ln(x) + 4x^(3/2)/3

Evaluating that at 5, then subtracting it evaluated at 3 gives you

x = 3ln(5/3) + 20sqrt(5)/3 - 4sqrt(3)

Or... 9.51

The second one:

The integral of 5 e^(-x/2) - 4 is

-10e^(-x/2) -4x

And evaluating it at 5 and 3, then subtracting them you get

x = -2(4e^(5/2) - 5e + 5)e^(-5/2)
or
x = -6.59


Hope that's in depth enough, if not I can clarify.

Answer 3

∫(3 to 5) ((3/x) + 2√x) dx
= [3 ln |x| + 2.x^(3/2)/(3/2)] [3 to 5]
= 3 ln 5 - 3 ln 3 + (4/3) (5^3/2 - 3^3/2)
= 3 ln (5/3) + (4/3) (5√5 - 3√3)

∫(3 to 5) (5e^(-x/2) - 4) dx
= [5e^(-x/2) / (-1/2) - 4x] [3 to 5]
= -10e^(-5/2) + 10e^(-3/2) - 20 + 12
= 10e^(-5/2) (e-1) - 8