(1)

1) Simplify with a substitution. 

It is often necessary to define a new variable , carefully chosen so that rewriting the integrand in terms of this new variable will make integration easier. In this case, the obvious variable to introduce will be defined by, 

                                                                      (2)

                                                                         (3)

Use equations (2) and (3) to rewrite (1).  

 2) Use integration by parts

To compute  use integration by parts. Ignore the constant out front for the moment, 

                                                 (4)

Define  and  and insert into the equation (4). 

 ,

                                               (5)

Let's factor the non-constant terms in equation (5), this will make the result easier to express when we convert back to 

We previously had a constant in front of the integral, 

Now we can write the integral terms of the original variable  by substituting equation (2) into the previous expression to obtain, 

To evaluate the integral, we perform the following substitution:

The derivative was found using the following rule:

Now, we rewrite the integrand and integrate:

The integral was performed using the following rule:

Finally, replace u with our original x term:

The Laplace Transform will be given by:

Since  when , we can change the integral to:

This is because when you change the lower bound of the integral, the exponent will only exist for values for which  is defined. 

Let 

This changes our integral to:

We can now move the  term out of the integral, which will give us:

To evaluate this integral, we first make the following substitution:

Differentiating this expression, we get:

Now, we can rewrite the original integral with our substitution and solve:

Finally, we have to replace u with our earlier definition:

Start by substituting  into the integral to get:

We can combine this into one large term:

Since .

This can only grow when: 

Let

We can now convert this back to a function of  by substituting , 

Add 2 and subtract 2 from the numerator of the integrand:.

Simplify and apply the difference rule:

Solve the first integral: .

Make the following substitution to solve the second integral:  

Apply the substitution to the integral: 

Solve the integral:

Combine the answers to the two integrals: .

Solution: 

We use substitution to solve the problem:

Let    and   

Therefore:

Here we use substitution to solve for the integrand.  Let u=sin(x) therefore du= cos(x)dx.  Plug your values back in:

To integrate this expression, you have to use u substitution. First, assign your u expression:

Now, plug everything back in so you can integrate:

Now integrate:

From here substitute the original variable back into the expression.

Evaluate at 2 and then 1.

Subtract the results: