Solve via integration by parts. Make the following substitutions:

    .

Plug in substitutions: .

Solve via integration by parts again. Make the following substitutions:

      .

Plug in substitutions: .

Simplify: .

Plug this integral back into our original equation:

Simplify: .

Add  to both sides of the equation: .

Factor  out of the right side of the equation: 

Divide both sides of the equation by : 

Factor out a sinx:

Apply Pythagorean identity to : .

Making the following substitutions:   

Apply substitutions: 

Solve integral: 

Convert u back to x: 

Make the following substitution:  .

Plug the substitution into the integrand: .

Factor out  from the denominator, and simplify: .

Apply Pythagorean identity to the denominator, and simplify:.

Solve integral:.

Use original substitution to solve for :        . 

Plug value for  back into solution for integral: .

This integral, , is a classic integration by substation problem. This is indicated by the presence of a composite function in which both a function and its derivative are present. 

For this we need to let our variable  , substituting this into our integral we produce the following:

and by substituting back in for u we find our final answer to be:
.

                                                    (1)

To compute this integral we will use a trigonometric substitution. First we need to do some algebra to write the integrated in a more suitable form, 

                                               (2)

Now we can apply a trigonometric substitution. 

                                                   (3)

                                         (4)

Notice we've chosen this substitution so that the radicand in equation (2) will conveniently reduce as follows, 

So now we have  , we need to convert back to , use equation (3), 

                                                       (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: