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Let $f$ be a continuous function defined on the interval $[2, \infty[$ such that $f(4)=14$, $|f(x)| < x^3+10$, and $\int_4^\infty f(x)e^{-x/4} dx=-5\;.$

Determine the value of: \int_4^\infty f\,'(x)e^{-x/4} dx\;.

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    You need to assume more than just continuity here. To use integration by parts (or the Fundamental Theorem of Calculus), you want $f(x)$ to be absolutely continuous.2012-03-18

3 Answers 3

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Hint: Try (definite) integration by parts:

\int_a^b f'(x) g(x) dx = \left[f(x) g(x)\right]_a^b - \int_a^b f(x) g'(x) dx.

Apply this to a suitable choice of functions $f(x), g(x)$ and bounds $a,b$. You should be able to find the right side of the equation using the conditions on $f$ provided by the question.


Solution: For completeness, let us see what we get. We apply the above to $f(x) = f(x), g(x) = e^{-x/4}, a = 4, b = \infty$. Finally we use e.g. the squeeze theorem to find the limit in the second line below to get the final result.

\begin{align} \int_4^{\infty} f'(x) e^{-x/4} dx &= \left[f(x) e^{-x/4}\right]_4^{\infty} - \int_4^{\infty} f(x) \frac{-1}{4} e^{-x/4} dx \\ &= \left[\lim_{x \to \infty} f(x) e^{-x/4} - f(4) e^{-4/4}\right] + \frac{1}{4} \left(\int_4^{\infty} f(x) e^{-x/4} dx\right) \\ &= 0 - \frac{14}{e} - \frac{5}{4} \end{align}

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Hint: Use integration by parts.

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Since it is given that

$\int_4^\infty f(x)e^{-x/4} dx=-5\;$

\begin{align*} f(x)e^{-x/4} dx &= -4 e^{\frac{-x}{4}} f(x) + 4 \int f'(x)e^{\frac{-x}{4}} dx \tag{A}\\ \end{align*}

In order to apply limits, you might have to think what to do next.

There is a reason why it is given that $|f(x)| < x^3+10 \hspace{5pt}$ does that mean that when you apply the limit as as $x \rightarrow \small{\infty}$ $\displaystyle{lim_{x \to \infty } e^{\frac{-x}{4}} f(x)} = {lim_{x \to \infty } \frac{f(x)} {e^{\frac{x}{4}}}} \rightarrow 0$ Therefore applying limits in $(A)$, you should get \frac{4f(14)}{e} + 4 \int f'(x)e^{\frac{-x}{4}} dx = -5 \Rightarrow \int_4^\infty f'(x)e^{\frac{-x}{4}} dx = -\frac{5}{4}-\frac{14}{e}
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    Missed dx in few places, so edited that2012-03-18