# Calculus 2 : Introduction to Integrals

## Example Questions

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### Example Question #23 : Fundamental Theorem Of Calculus

Explanation:

This is a Second Fundamental Theorem of Calculus problem.  Since the derivative cancels out the integral, we just need to plug in the bounds into the function (top bound - bottom bottom) and multiply each by their derivative.

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### Example Question #24 : Fundamental Theorem Of Calculus

Differentiate:

Explanation:

Differentiate:

Use the Second Fundamental Theorem of Calculus:

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If the function  is continuous on an open interval  and if  is in , then for a function  defined by , we have

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To understand how to apply the second FTOC, notice that  in our case is a function of a function. The chainrule will therefore have to be applied as well. To see how this works, let . The function can now be written in the form:

Now we can go back to writing in terms of , the derivative is,

### Example Question #25 : Fundamental Theorem Of Calculus

In exponentially decaying systems, often times the solutions to differential equations take on the form of an integral called Duhamel's Integral. This is given by:

Where  is a constant and  is a function that represents an external force.

Equations of the form  are  sinusoidal functions, where  is the imaginary constant. Determine the external force  needed to produce solutions  of sinusoidal behavior

Explanation:

We plug in  into our formula and determine the external force .

Taking the derivative of each side with respect to , and applying the fundamental theorem of calculus:

Solving for ,

### Example Question #26 : Fundamental Theorem Of Calculus

In harmonic systems, often times the solutions to differential equations take on the form of an integral called Duhamel's Integral. This is given by:

Where  is a constant and  is a function that represents an external force.

What external force is needed in order to obtain  if

Explanation:

In order to solve this, we substitute  into our equation and solve for :

Taking the derivative of each side with respect to , we get that:

Using the fundamental theorem of calculus:

Solving for :

Since ,

Let

Find .