By the fundamental theorem of calculus:

First compute the indefinite integral. Remember that

Evaluate the integral from the limits given

is alwaysand any number to the power of 0 is always 1.

therefore:

上的积分都是不当的,因为李mit at infinity. To handle this, we use the theorem that if an integral

exists, then it is equal to the value of

.

一个pplying this to our given integral, we get:

Using the power rule for integration, which is valid since,we obtain:

Here the second term in the limit didn't depend on, so it evaluates to

.

To evaluate the limit for the first term, we note that from, we have that.

Therefore the power onis always less thanfor any. We have the theorem that

.

Therefore the first term in the limit asvanishes, and we are left with the result

.

The integral we are given is improper due to the upper bound being infinity. To integrate, we must rewrite the integral with a limit for the upper bound:

Note that we moved the constant outside of the integral and the limit.

Now, we can integrate, remembering to keep the limit in front ofeverystep in the process:

The integral was performed using the following rule:

Now, we can definitely integrate by plugging in the upper and lower bounds and subtracting the results:

Note that the absolute value goes away because the bounds of the integral are both positive.

The function goes to infinity as the natural log of infinity is infinity.

Substitute a in for infinity and solve the integral:

Now, take the limit of this expression as a approaches infinity:

To evaluate the improper integral (infinity is its upper limit of integration which makes it improper), we must use a limit for the upper bound.

But before we do this, we can use a u-substitution to rewrite the integral as something we can easily integrate:

,

The derivative is found using the identical rule to the derivative.

Now, rewrite the integral, with a limit for the upper bound, and integrate

The integral was found using the following rule:

Next, we can replace u with our x terms, and perform the definite integration:

一个s t goes to infinity, the denominator of the first term goes to infinity, which means the entire term goes to zero. We are left with

To evaluate the improper integral (infinity is its upper bound), we must use a limit on the upper bound:

Note that we are approaching infinity from the right because it is the upper bound of integration.

We can now integrate (don't forget the limit!):

The following rule was used for integrating:

Now, the definite part of the integration:

Note that because the bounds of the integral were both positive, we removed the absolute value sign. When t goes to infinity, the natural logarithm function goes to infinity. The natural logarithm of 1 is zero, so we are left with infinity.

This integral is improper because the lower bound creates a zero in the denominator. To integrate, we must use a limit:

The following rule was used:

Now, the definite part of the integral:

The natural logarithm of 1 equals zero, and as the natural logarithm approaches zero, the function goes to negative infinity. The negative sign in front of the function makes it go to infinity.

We can evaluate this integral using integration by parts

. (Start)

(Use integration by parts, with)

.