Study questions...need to know how for exam

**jahichuanna** · June 14th 2009, 06:16 PM

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

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

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

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

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

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

Thank you!!!

**apcalculus** · June 14th 2009, 08:31 PM

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

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

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

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

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

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

Thank you!!!

I will try to help with some of these.

1) Apply the change of base property.
$\int^5_1 \frac{log_5 x}{x}dx =\int^5_1 \frac{1}{ln 5}\frac{ln x}{x}dx$
Pull the $\frac{1}{ln5}$ in front of the integral sign.
Use the fact that the first derivative of ln x is $\frac{1}{x}$ to integrate.
5) $\int x^3e^xdx$
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!

**HallsofIvy** · June 15th 2009, 03:40 AM

Are you and jangalinn in the same course?

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

**jahichuanna** · June 15th 2009, 04:02 AM

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

**AMI** · June 15th 2009, 04:44 AM

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

**apcalculus** · June 15th 2009, 06:05 AM

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!)

(misunderstood HallsofIvy)

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