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# Establish the integration formula for arccot x

Establish that the following integration formula is correct:

Proof. We can establish this formula using integration by parts. Let

where we established the formula for the derivative of in this exercise (Section 6.22, Exercise #3). Then we have

# Prove integration formulas for eaxcos (bx) and eaxsin (bx)

Let and be constants with at least one of them nonzero and define

Using integration by parts, establish the following formulas for constants ,

Using these formulas prove the following integration formulas,

To establish the formula we use integration by parts letting

Then we can evaluate using the formula for integration by parts,

To establish the second formula, , we use integration by parts again. Let

Then we have

This establishes the two requested equations, now we prove the two integral identities.

Proof. Solving for in the second equation above we have

Plugging this into the first equation we have

Next, for the second integral equation we are asked to prove, we use the formula we obtained for above,

Then, we use the expression we obtained for into this,

This implies,

# Evaluate the indefinite integral of ex1/2

Compute the following indefinite integral

To evaluate this integral we want to make a substitution. First multiply the numerator and denominator by to obtain,

Now define the function by

This implies

Therefore, using the method of substitution, we have

# Evaluate the indefinite integral of x2e-2x

Compute the following indefinite integral,

To compute this integral we will integrate by parts twice. First, let

Therefore we have

To evaluate this next integral we use integration by parts a second time with

Giving us

So, putting this back into our formula we have

# Evaluate the indefinite integral of x2ex

Compute the following indefinite integral,

We compute the integral using integration by parts. Let,

Then we have

But we know from a previous exercise (Section 6.17, Exercise #13) that

Therefore we have,

# Evaluate the indefinite integral of xe-x

Compute the following indefinite integral,

To evaluate this integral we use integration by parts, letting

Therefore,

# Evaluate the indefinite integral of xex

Compute the following indefinite integral:

To evaluate this integral we use integration by parts, letting

Then we have,

# Prove a recursion formula for the integral of xm logn x

Prove the following recursion formula holds:

Apply this formula to find the solution of

Proof. To obtain this recursion formula we proceed by integrating by parts. To that end, let

Then we have

Now, to find the solution of

we apply the formula with . This gives us,

Next we apply the formula to the resulting integral with and . Therefore we have

Finally, we apply the formula to the integral above with and to obtain,

# Find the integral of xn log (ax)

Evaluate the following integral:

Here we want to use integration by parts. To that end we define,

if .

Therefore, integrating by parts we obtain the integral for all .

Now, for the case that we have

Since is the derivative of we make the substitution and to obtain

# Find the integral of x log2x

Evaluate the following integral:

To evaluate this integral we use integration by parts, defining

Then we have

From the previous exercise (Section 6.9, #18) we know

Therefore,