The inverse of the exponential function

dalterego
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The inverse of the exponential function...

Homework Statement



Find the inverse of the function = e ^ (x^3)

Homework Equations



The inverse of the exponential function = the natural logarithm of that same function

The Attempt at a Solution



inverse of f(x) = ln(x^3) ?

This doesn't seem right.
 
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Nope. Let y=e^(x^3). Take Ln of both sides and express x in terms of y. Then replace y with x and you're done.
 


Alright, so is the answer

inverse of the function = (ln x)^1/3 ?
 


Yeah.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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