Question

How do you integrate `int x^3 sin x dx` using integration by parts?

Answer 1

For the integration of a product of two functions (First function)*(Second function) = `f(x)`,

`int f(x)*dx = (First function)*int (Second function)*dx - int (d/dx(First function)*int(Second function)*dx`.

This is called integration by parts.

Explanation:

The choice of first function and second function is arbitrary in case of most functions.

Here, we have `f(x) = x^3*Sin x` and we will choose `x^3` as the first function and `Sin x` as the second.

Thus, `int f(x)*dx = int x^3*Sin x*dx`

`implies int f(x)*dx = x^3 int Sin x*dx - int (d(x^3)/dx*int Sin x*dx)*dx`

`= -x^3*Cos x + int 3x^2*Cos x*dx`

`= -x^3*Cos x + 3[x^2int Cos x*dx - int (d/dx(x^2) intCos x*dx)*dx]`

`= -x^3*Cos x + 3[x^2Sin x - int 2xSin x*dx]`

`= -x^3*Cos x + 3[x^2Sin x - 2(x int Sin x*dx - int (d/dx(x)int Sin x*dx)*dx]`

`= -x^3Cos x + 3[x^2Sin x - 2(-xCos x + int Cos x*dx)]`

`= -x^3Cos x + 3[x^2Sin x + 2xCos x - 2Sin x]`

Since it is an indefinite integral, we add an arbitrary constant to it.

`int x^3Sin x*dx = - x^3Cos x + 3x^2Sin x + 6xCos x - 6Sin x + C` where `C` is the integration constant.