Integration by Parts: Solving ( int x^3 e^x sin x ,dx )

Integration by parts is a method derived from the product rule of differentiation to solve integrals of the form ( int u , dv ). The method can be expressed as:

Formula for Integration by Parts

Given ( int u , dv ), the integral can be rewritten as:

[ int u , dv uv - int v , du ]

Problem Statement and Solution Approach

Let's solve the integral ( I int x^3 e^x sin x ,dx ). To solve this, we need to break it down into smaller, more manageable integrals.

Solving ( I_1 int e^x sin x ,dx )

First, we focus on the integral ( I_1 ).

Step 1: Applying Integration by Parts

We choose ( u e^x ), so ( du e^x , dx ) and ( dv sin x , dx ), which gives ( v -cos x ).

Using the formula for integration by parts:

[ I_1 -e^x cos x - int -cos x , e^x , dx ]

[ implies I_1 -e^x cos x int e^x cos x , dx ]

We will now solve ( int e^x cos x , dx ) using integration by parts.

Step 2: Solving ( I_2 int e^x cos x ,dx )

We choose ( u e^x ), so ( du e^x , dx ) and ( dv cos x , dx ), which gives ( v sin x ).

Using the formula for integration by parts:

[ I_2 e^x sin x - int sin x , e^x , dx ]

We can see that the integral ( int sin x , e^x , dx ) is the same as the one we had in ( I_1 ), but with a negative sign.

So, combining both parts:

[ 2I_1 e^x sin x - cos x ]

[ implies I_1 frac{1}{2} e^x sin x - frac{1}{2} cos x ]

Solving ( int x^3 e^x sin x ,dx )

Now we return to the original integral ( I int x^3 e^x sin x ,dx ).

Step 3: Applying Integration by Parts

We choose ( u x^3 ), so ( du 3x^2 , dx ) and ( dv e^x sin x , dx ), which gives ( v frac{1}{2} e^x sin x - frac{1}{2} cos x ).

Using the formula for integration by parts:

[ I x^3 left( frac{1}{2} e^x sin x - frac{1}{2} cos x right) - int left( frac{3}{2} x^2 e^x sin x - frac{3}{2} x^2 e^x cos x right) , dx ]

This integral now involves ( int x^2 e^x sin x , dx ) and ( int x^2 e^x cos x , dx ), which can be solved by applying integration by parts again.

Final Answer

The integral ( I int x^3 e^x sin x ,dx ) can be expressed as:

[ I x^3 left( frac{1}{2} e^x sin x - frac{1}{2} cos x right) - int left( frac{3}{2} x^2 e^x sin x - frac{3}{2} x^2 e^x cos x right) , dx ]

Conclusion

Integration by parts is a powerful tool for solving integrals involving products of functions. By breaking down the integral into smaller, more manageable parts, we can solve complex integrals using the principles of integration by parts.

Related Keywords

Integration by parts, solving integrals, mathematical techniques