1. ## differentiation

1, find s in terms of t if ds/dt=3t-8/tsquared and s=1and1/2 when t= 1

1, square root of (2-3x)dx

2, given that dy/dx=3/x-2 and y =10 when x=0, find an expression for y in terms of x

1, find( 2e^3x-e^-x) dx
Thanks!

2. Originally Posted by afan17
1, find s in terms of t if ds/dt=3t-8/tsquared and s=1and1/2 when t= 1

1, square root of (2-3x)dx

2, given that dy/dx=3/x-2 and y =10 when x=0, find an expression for y in terms of x

1, find( 2e^3x-e^-x) dx
Thanks!

The first one is, I presume, to solve
$\frac{ds}{dt} = 3t - \frac{8}{t^2}$, $s(1) = \frac{3}{2}$

This equation is separable:
$ds = \left ( 3t - \frac{8}{t^2} \right )~dt$

Upon integrating
$s = \frac{3}{2}t^2 + \frac{8}{t} + C$

We know that s(1) = 3/2, so
$\frac{3}{2} = \frac{3}{2} \cdot 1^2 + \frac{8}{1} + C$

Solve for C.
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For the second one I presume you are trying to integrate?
$\int \sqrt{2 - 3x}~dx$

Let $y = 2 - 3x \implies dy = -3~dx$

$\int \sqrt{y}~\frac{dy}{-3}$
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For the third I presume this is
$\frac{dy}{dx} = \frac{3}{x - 2}$, $y(0) = 10$

This is, again, a separable equation:
$dy = \frac{3}{x - 2}~dx$

Integrate both sides by using the substitution u = x - 2.
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I presume the last one is an integration as well and is
$\int (2e^{3x}-e^{-x})~ dx$
$\int 2e^{3x}~dx - \int e^{-x}~dx$