Find the derivative of an integral

In summary, the First Fundamental Theorem of Calculus can be used to easily solve the integral of sin(t) with respect to t. The result is cos(1) - cos(x). Alternatively, one can differentiate the integral with respect to x, resulting in the function sin(x). Remembering the FTC can make this process simpler.
  • #1
IntegrateMe
217
1
[tex]\frac {d} {dt}\int_{1}^{x} sint dt[/tex]
 
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  • #2
This one is easy if you remember the First Fundamental Theorem of Calculus.
 
  • #3
Thanks, i guess i was just looking for what particular method to use.
 
  • #4
Well, there are two methods...

You could solve the integral, which isn't too hard (the answer is cos(1)-cos(x)), and then differentiate it wrt x to reveal the function. But it's easier to just remember the FTC.
 
  • #5
The result is zero.
 
  • #6
IntegrateMe said:
[tex]\frac {d} {dt}\int_{1}^{x} sint dt[/tex]

Are you sure that you notated this correctly? It looks unusual to me. The integration would be done with respect to a variable t (essentially a dummy variable), and the result will be a function of x (the upper limit). The derivative of this with respect to t would then be zero.

Did you intend the derivative to be with respect to x?
 
  • #7
Yes, sorry, it's d/dx

Wouldn't the result be just "sinx" ?
 
  • #8
IntegrateMe said:
Yes, sorry, it's d/dx

Wouldn't the result be just "sinx" ?
yes.
 
  • #9
Thank you! :)
 

What is the purpose of finding the derivative of an integral?

The derivative of an integral is used to calculate the rate of change of a quantity over time. This can be useful in many fields such as physics, economics, and engineering.

How do you find the derivative of an integral?

The derivative of an integral can be found by using the Fundamental Theorem of Calculus, which states that the derivative of an integral is equal to the original function being integrated.

What is the difference between the derivative and the integral?

The derivative and integral are inverse operations. The derivative calculates the rate of change of a function, while the integral calculates the area under the curve of a function.

What are some common techniques for finding the derivative of an integral?

Some common techniques for finding the derivative of an integral include using the power rule, the chain rule, and substitution. It is important to also be familiar with the properties of derivatives and integrals.

Why is it important to find the derivative of an integral?

It is important to find the derivative of an integral because it allows us to analyze the behavior of a function and make predictions about its future values. It also has many practical applications in fields such as physics, engineering, and economics.

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