Study questions...need to know how for exam

I have a set of problems that I don't know how to do. I'm using them to study for my exam, so please show a step-by-step or at least a basic idea how to do it.

1) $\displaystyle \int^5_1 \frac{log_5 x}{x}dx$

2) Evaluate: $\displaystyle lim_{x \to 0} (\frac{1}{ln(x+1)} - \frac{1}{x})$

3) Californium-252 has a half-life of 2.645 years. If the equation describing the decay of the readioactive nuclei is given by $\displaystyle y=y_0e^{kt}$, find:
a) the value of k
b) how long it will take for 95% of the radioactive nuclei to disintegrate.

4) Solve:$\displaystyle \frac{dy}{dx} + y = x$
I don't even know what I'm supposed to do on this one...(Wondering)

5) $\displaystyle \int x^3e^xdx$

6) Use $\displaystyle \frac{dy}{dx} = ky(M-y)$ with k=.001 and M=150 to write y as a function of t. $\displaystyle (y_o = 5)$

Thank you!!!

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Originally Posted by jahichuanna

I have a set of problems that I don't know how to do. I'm using them to study for my exam, so please show a step-by-step or at least a basic idea how to do it.

1) $\displaystyle \int^5_1 \frac{log_5 x}{x}dx$

2) Evaluate: $\displaystyle lim_{x \to 0} (\frac{1}{ln(x+1)} - \frac{1}{x})$

3) Californium-252 has a half-life of 2.645 years. If the equation describing the decay of the readioactive nuclei is given by $\displaystyle y=y_0e^{kt}$, find:
a) the value of k
b) how long it will take for 95% of the radioactive nuclei to disintegrate.

4) Solve:$\displaystyle \frac{dy}{dx} + y = x$
I don't even know what I'm supposed to do on this one...(Wondering)

5) $\displaystyle \int x^3e^xdx$

6) Use $\displaystyle \frac{dy}{dx} = ky(M-y)$ with k=.001 and M=150 to write y as a function of t. $\displaystyle (y_o = 5)$

Thank you!!!

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I will try to help with some of these.

1) Apply the change of base property.
$\displaystyle \int^5_1 \frac{log_5 x}{x}dx =\int^5_1 \frac{1}{ln 5}\frac{ln x}{x}dx$
Pull the $\displaystyle \frac{1}{ln5}$ in front of the integral sign.
Use the fact that the first derivative of ln x is $\displaystyle \frac{1}{x}$ to integrate.
5) $\displaystyle \int x^3e^xdx$
Integration by parts
u=x^3 u'=3x^2
v'=e^x v=e^x
You will need to integrate by parts three times here. In the next stage set u=x^2, and in the one after that set u=x.

6. Separate variables, all the y terms and dy on one side, and k dx on the other. You will need to use partial fractions on the y-side.


It would be easier if you posted one problem at a time.
Perhaps other folks will offer hints to the other problems. Good luck!

Are you and jangalinn in the same course?

(Perhaps you are both taking the same course but seldom actually in the class!)

nah....we don't have AP classes at our school, just the test. so we can't be.

4)You have to solve the affine diff. eq. $\displaystyle \frac{dy}{dx}=-y+x$
$\displaystyle \bullet$ first solve the associated linear eq: $\displaystyle \frac{d\overline y}{dx}=-\overline y$.
The solution is $\displaystyle \overline y(x)=c\exp(-x)$, where $\displaystyle c\in\mathbb{R}$ is a constant.
$\displaystyle \bullet$ search for solutions (of the initial eq) in the form $\displaystyle y(x)=c(x)\exp(-x)$:
Put that in $\displaystyle \frac{dy}{dx}=-y+x$ and you obtain $\displaystyle c^{\,\prime}=x\exp(x)\Rightarrow c(x)=(x-1)\exp(x)+C\Rightarrow y=(x-1)+C\exp(-x)$, where $\displaystyle C\in\mathbb{R}$ is a constant.

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Originally Posted by HallsofIvy

Are you and jangalinn in the same course?
(Perhaps you are both taking the same course but seldom actually in the class!)

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(misunderstood HallsofIvy)