Calculus 2 : Definite Integrals

Example Questions

Example Question #191 : Definite Integrals

If , and  what is

Explanation:

To find the total integral on our interval from  to , we need to find the area under each subinterval and add them all up.  The first and last part are given in the problem statement, but the middle interval has backwards limits.

Remember:

Therefore, we need to switch the sign on the middle area.  Therefore,

Example Question #192 : Definite Integrals

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.  represents the overall solution to the differential equation.

Determine  if the external force is given by

Explanation:

We can rewrite the integral as

Using  substitution, we let

We can therefore write the integral as:

Since .

Example Question #193 : Definite Integrals

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.

Determine the value at  if

Explanation:

Plugging  into the equation:

Integrating by parts, we can assign values for , and :

Plugging into our equation:

The first term becomes:

The integral term becomes

Plugging everything else in we get:

Example Question #191 : Definite Integrals

Evaluate the following definite integral:

Explanation:

This integral requires use of the power rule for antiderivatives, which simplifies as follows:

Example Question #195 : Definite Integrals

Evaluate the following definite integral:

Explanation:

Make the substitution:

Where

The limits of the integral, 0 and 1, are also changed in the substitution:

Using this in the original expression:

Example Question #196 : Definite Integrals

Evaluate the following definite integral:

Explanation:

To solve this with integration by parts, we rewrite the expression in the form

where

and

To integrate, apply the formula for integration by parts:

Example Question #197 : Definite Integrals

Evaluate the following definite integral:

Explanation:

This integral can be solved by using partial fractions.  First, we have to factor the denominator write the fraction as a sum of two fractions:

Next, we can solve for A and B:

When we let x=3:

When x=2:

Replacing A and B in the integral, we can now solve it:

Example Question #198 : Definite Integrals

Evaluate the following integral:

Explanation:

Recall that given two real numbers b and a with  the integral  is equivalent to .

Therefore,

is equivalent

.

Evaluating the integral one finds