Extremely hard calculus problem...?

Question

See if you can get it...

Find dy/dx of the integral of sin(t^2), on the interval [0,sqrt of x]

it is sin(t^2) not x^2!

Answer 1

f(x) = Integral (sin (t^2) dt ) {evaluated from 0 to sqrt(x)}

And you want to find f'(x).

These types of questions are known as the Fundamental Theorem of Calculus questions. Here's the clincher: They are EXTREMELY easy, but are covered in Calculus II so briefly that many students lose a lot of marks on this.

First off: there is absolutely NO differentiation or integration involved!! The only thing that's involved is using the chain rule.

When you solve f'(x), all you have to do is plug in every instance of the upper boundary for t, apply the chain rule, and then plug in the lower boundary for t, and apply the chain rule for that.

f'(x) = sin ( [sqrt(x)]^2 ) {1/[2sqrt(x)]} - sin(0^2) {0}

I used squiggly brackets to denote the use of the chain rule. This then of course simplifies into

f'(x) = sin (x) (1/[2sqrt(x)])

As you can see:
ZERO integration, ZERO differentiation. All there was to it was substitution, and application of the chain rule.

I'll give you another example:

EXAMPLE:
f(x) = Integral ( sqrt(1 + t^3) dt ) {evaluated from sin(x) to e^x}

Solving f'(x), remove the integral symbol, plug in sin(x) for every occurrence of t, apply the chain rule to sin(x), MINUS, plug in e^x for every occurrence of t, apply the chain rule to e^x.

f'(x) = [sqrt(1 + [sin(x)]^3)] {cos(x)} - [sqrt(1 + (e^x)^3)] {e^x}

As you can see, there was no need to solve for the integral of
sqrt(1 + x^3) (which is a very difficult integral anyway).

How you identify these types of questions are:
(1) It involves taking the derivative of an integral expressed in terms of t and
(2) There is a function of x in the bounds for integration.

EXAMPLE #2:

f(x) = Integral ( e^(t^2) dt ) {evaluated from ln(x) to x^3}

f'(x) = e^( [ln(x)]^2 ] {1/x} - e^[ (x^3)^2 ] {3x^2}

Hopefully you get it now and show everyone else this. Like I said earlier, this is covered in such a TINY part of the course, yet when it appears on an exam, students get stumped. Don't let this happen to you.

Answer 2

That is so ironic. The guy who asked a q b4 u titled his Q "simple calculus problem"

Answer 3

it would be the sin((x^1/2)^2) *1/2x^(-1/2) unless I am much mistaken