Extension One Revising Game (1 Viewer)

jet · May 31, 2009

Typical one:
$\text{Integrate } f(x) = 3sin^2 (4x) \text{ with respect to } x$

gurmies · May 31, 2009

$\cos8x=1-2\sin^{2}4x \\\\ \sin^{2}4x=\frac{1}{2}(1-\cos8x) \\\\ 3\int \sin^{2}4xdx=\frac{3}{2}\int (1-\cos8x) \\\\ = \frac{3}{2}x-\frac{3}{16}\sin8x+c$

New question:

$Show \,\, that \,\, \int \frac{1}{x\sqrt{x^{2}-1}}dx=\cos^{-1}(\frac{1}{x})+c \,\, using \,\, the \,\, substitution \,\, x=\sec\theta \\\\ Hence \,\, show \,\, that \,\, \int_{2}^{3}\frac{1}{x\sqrt{x^{2}-1}}dx=\cos^{-1}(\frac{1+2\sqrt6}{6})$

shaon0 · May 31, 2009

gurmies said:
$\cos8x=1-2\sin^{2}4x \\\\ \sin^{2}4x=\frac{1}{2}(1-\cos8x) \\\\ 3\int \sin^{2}4xdx=\frac{3}{2}\int (1-\cos8x) \\\\ = \frac{3}{2}x-\frac{3}{16}\sin8x+c$

New question:

$Show \,\, that \,\, \int \frac{1}{x\sqrt{x^{2}-1}}dx=\cos^{-1}(\frac{1}{x})+c \,\, using \,\, the \,\, substitution \,\, x=\sec\theta \\\\ Hence \,\, show \,\, that \,\, \int_{2}^{3}\frac{1}{x\sqrt{x^{2}-1}}dx=\cos^{-1}(\frac{1+2\sqrt6}{6})$

Let x= sec@, dx=sec@tan@ d@
I
= S (sec@tan@/(sec@sqrt((sec@)^2-1))) d@
= S d@
= @ + C
= arccos(1/x) +C

For I with limits x=3,2.
I = arccos(1/3)-arccos(1/2)
Let a=arccos(1/3), b=arccos(1/2)
I = a-b
cos(I) = cos(a-b)
cos(I) = cos(a)cos(b)+sin(a)sin(b)
cos(I) = (1/3)(1/2)+(sqrt(8)/3)(sqrt(3)/2)
cos(I) = (1/6)+sqrt(24)/6
I = arccos((1+2sqrt(6)/6)

Someone can ask a question on my behalf.

Trebla · May 31, 2009

Try this:

$$Consider the differential equation$\\\\\dfrac{d^2y}{dx^2}+2\dfrac{dy}{dx}-3y=0$ With conditions $ x = 0$ at $y=2$ and $\dfrac{dy}{dx}=-2 \\\\$(i) Find values of $k$ such that $y = e^{kx}$ satisfies this equation$\\\\$(ii) The general solution of this equation is $y = Ae^{px}+Be^{qx}$ for some constants $p$, $q$, $A$ and $B$. Find the values of these constants.$

kurt.physics · Jun 11, 2009

As no one has been able to solve Trebla problem, then (if you dont mind Trebla), for the sake of furthering the progress of this thread, i will post a new question.

given $f(x) = \sin\left(\omega x + \phi \right)$ where $\omega > 0$ and $\left|\phi \right| < \frac{\pi}{2}$

if:

$\cos\left(\frac{\pi}{2}\right) \cos\left(\phi\right) - \sin\left(\frac{3\pi}{4}\right) \sin\left(\phi\right) = 0$

then find the value of $\phi$

addikaye03 · Jun 11, 2009

Trebla said:
Try this:

$$Consider the differential equation$\\\\\dfrac{d^2y}{dx^2}+2\dfrac{dy}{dx}-3y=0$ With conditions $ x = 0$ at $y=2$ and $\dfrac{dy}{dx}=-2 \\\\$(i) Find values of $k$ such that $y = e^{kx}$ satisfies this equation$\\\\$(ii) The general solution of this equation is $y = Ae^{px}+Be^{qx}$ for some constants $p$, $q$, $A$ and $B$. Find the values of these constants.$

Haven't attempted ii) yet but is the answer to i) k=1 and -3/2

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Extension One Revising Game (1 Viewer)

jet

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