Integration help please!!!!!

Question

mathmusician · Answer

Compute the derivative: $$\frac{ d }{ dx}\int\limits_{cosx}^{sinx}\sqrt{t}dt$$

mathmale · Answer

The easiest way to do this is to convert radical form sqrt (t) to exponential form:

t^(  ?  )

or
$$t^?$$

mathmale · Answer

Then apply the power rule to integrate your result.

Try it, please.

mathmusician · Answer

Okay it would be $$t ^{1/2}$$

mathmusician · Answer

you want me to find the anti-derivative?

mathmale · Answer

Yes.  What would be$$\int\limits_{a}^{b}t ^{1/2}dt?$$

mathmale · Answer

Yes.  this is the "antiderivative" or "definite integral" of sqrt (t) from t=a to t=b.

mathmusician · Answer

anti-derivative and definite integral are not the same thing you are thinking of the indefinite integral

mathmusician · Answer

when the integration equation has the values of a and b the it is a definite integral

mathmale · Answer

"antiderivative" is generally similar to "indefinite integral."  

Your posted problem is a definite integral, so my asking you to integrate t^(1/2)  on the integral [a,b], that's a definite integral and is appropriate.

mathmusician · Answer

oh okay

mathmale · Answer

I politely disagree with your most recent statement.  Whether the limits of integration are and b or   23 to 45, that's a definite integral you're dealing with.

You're arguing with a math teacher with 43 years of experience.

mathmusician · Answer

i dont know the interval so i cant evaluate the definite integral

mathmusician · Answer

unless there is certain way to do it

mathmale · Answer

You do know the limits of integration.  They are from cos x to sin x.

mathmale · Answer

At this point we have to switch to a different railway track.
What you have posted is "a function defined as an integral."
You can actually integrate sqrt (t) and use cos x and sin x as the limits of integration, OR

you could use a more sophisticated method:

When a function f(x) is definited as the (unfinished) integral of some other function,
finding the value of the derivative is a snap.

mathmusician · Answer

the anti-derivative of t^(1/2) is $$\frac{ 2t ^{\frac{ 3 }{ 2}} }{ 3 }+C$$

mathmusician · Answer

Sorry for arguing with you up there

mathmale · Answer

Yes.  F(b) - F(a) would then be found by letting a=cos x and b=sin x.  The constant C will disappear, as is true in the case of all definite integrals.

Argue, argue.

;)

mathmusician · Answer

sinx - cosx = my answer?

mathmusician · Answer

wait it would be $$\sqrt{sinx}-\sqrt{cosx}$$

mathmusician · Answer

because of f(b)-f(a)

mathmale · Answer

If you are to evaluate the following:
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