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I'm having trouble with these two calculus exercises. The first one is a simple case of Integral differentiation. I have to differentiate:

$$\int_x^b \frac{x}{1+t^2+\cos(t^2)}dt$$

But sincerely I have no clue. I've been tempted to switch the limits of integration and put a minus in front of the integral, but that doesn't help me at all. What is that $x$ doing in that fraction?

The other one states:

Find $f^{(n)}(x)$ of the function $\frac{x^n}{1-x}$.

I know I have to find the nth derivative of the function but, as $n$ is present in the function, I'm confused. Does this mean that, if $n = 1$, I have to find the first derivative of $\frac{x^1}{1-x}$, if $n = 2$ find the second derivative of $\frac{x^2}{1-x}$ and so? I think that doesn't even make sense.

Thanks.

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Are you trying to differentiate the integral once with respect to $x$? –  Mark Bennet May 21 '12 at 21:18
    
Yes, that's what I'm trying to do. –  Teckizt May 21 '12 at 21:20
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For first question, depends how much machinery you want to use. But your function is a product $-x\int_b^xf(t)\,dt$. (I have used your interchange idea.) Now use product rule, and stuff about differentiation of an integral. –  André Nicolas May 21 '12 at 21:20
    
Hi Teckizt. You have to notice a pattern for the derivative part. First derivative is $1/(x-1)^2$. The third is $((3-2x)x^2)/(x-1)^2$. The four is $((4-3x)x^3)/(x-1)^2$. Notice the pattern? –  Nico Bellic May 21 '12 at 21:23
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1 Answer 1

up vote 2 down vote accepted

$$ \int_x^b \frac{x}{1+t^2+\cos(t^2)}dt = x \int_x^b \frac{1}{1+t^2+\cos(t^2)}dt. $$ The thing above works because the $x$ that got pulled out does not change as $t$ goes from $x$ to $b$. In other words, it's a "constant".

"Constant" always means not depending on something. What the something is, may depend on context.

Next you can use (1) the product rule, and (2) $\displaystyle\frac{d}{dx}\int_\text{whatever}^x f(t)\,dt = f(x)$ (and of course first swicth the bounds and put the minus sign there).

Your interpretation of the second question is correct. The exponent will be the same as the number of times you differentiate. You might want to try it for $n=1$, then for $n=2$, then for $n=3$, and see if a pattern emerges. It may be that you will want to prove the pattern by mathematical induction.

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