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Evaluate the followimg WO integrsle:ec2(*c) &0}00s$ lrlgi #) d...

Question

Evaluate the followimg WO integrsle:ec2(*c) &0}00s$ lrlgi #) d

Evaluate the followimg WO integrsle: ec2(*c) & 0} 00s$ lrlgi #) d



Answers

$$\text { Evaluate } \frac{d}{d x}\left(x^{e}+e^{x}\right)$$

So the problem for here is to show how FX quote, which is equal to G of X to the power of H of X, can be rewritten in a form using natural logs and natural base meaning E. So we start by taking the natural log of both sides. True log that's of X equals natural log of G of X to the H of X, and by our rules of logarithms, you take our h of X in place it in front of natural lock So we could h of x front, and I will rewrite this. Sinbad is equivalent to h of x times the natural log of G. Max. So we need to This is still the natural log, the athlete axe. So we need to bring it back out. So to move the natural log of f of X, we raised every make this the exponents to e. So we get e to the natural log that fax e to the h of X. How does the natural g of X and this just becomes on the left side? E to the natural log of anything is equal to the whatever is in the natural walk. So we're just left with half of X. We left just equivalent to yeah, to the h of x times, the natural log G blacks. And to my knowledge, this does not simplify any further. So that is how it could rewrite it using natural logs and the natural base.

Here we are asked to integrate an exponential function. So first of all, we'll substitute three x s u. So that would give us three. B X is equal to deal. So which means the X is equal toe. Do you buy three? So we already have the X on the right side. So that would give us lead by three. So we can rewrite this race to you. Do you buy three So that would be won by three integral areas to you, Do you? So we already know that integral areas two x dx is areas two X plus c where C is integration constant. It can use this formula here that would give us one by three he raised to you. Let's see, we can substitute the value of you which we have already substitute here, which is three X and that will give us one by three years to three x less see, So this would be the final answer. We can even a reject this thing by differentiating in the answer and going back to the question. So d by the X off he raised toe one by three he raised to three x plus C that would give us one by three. Differential of erased 23 exists erased 23 X in tow by chain rule, or differential of three X, that is three plus differential of a constant is zero. So by canceling the three, we'll be getting a raise to three x, which is the question which was asked, So this is the right answer to this question.

So here we're having an exponential function, Toby Integrator. So we're taking expired three As you species Absolutely. So x by three is equal to you. That will give us BX by three is equal to you. But we have only dx on the integration. So we'll keep it be X is equal to three into deal. So that would give us a necessary resolution. So that would give us he raised to you the into three deal. So we'll be taking three outside. That would be three integral areas to you. Deal. So we know the formula for integral erase too. Xbox s here is to express integration. Constancy. So we need justice formula to do this question. So this would give us three. Here s to you, plus see which is the integration conscience. But this is common. Three is common to both in the Asian continent area street so that by re substituting the value of you which is expired three and they give us three years to x by three plus three c eso again we can simplify. There's three years expired three plus c dash, where we're substituting three CSC rash which is another constant So this is the final answer. So we can even reach ticket by differentiating the the answer to get back to the question. So do you buy the eggs off? Three areas two x by three Let's see rash. So that will give us three areas two x by three into derivative of X by trees one by three because of gentle and derivative of constant is zero. So by canceling three would get a raise to exploit three. So this is we're back to the question from the answer. So this is the right answer which is areas two x by 30 or we can call it s another constancy.

Officially call that the derivative of hyperbolic attention that people too hyperbolic sequence where w so we can use use up We'll let you people to report conscience of W and into you that people too hyperbolic Deakins squared of the view of you So this portion that's equal to be you So we have the integral of you you which is equal to use square over to see Looking in back you we get type of all tangents w squared over two plus t.


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