Compute the following indefinite integrals with the use of the integration by parts method. Show steps. ∫ x sin x dx

The integral of (xsinx)dx=sinx-xcosx+C.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

We are asked to perform `int xsinxdx` using integration by parts.

The concept of integration by parts comes from taking the derivative of a product. Recall that the derivative of a product is not the product of the derivatives. Instead, given differentiable functions f and g, `d/(dx)(fg)=f'g+fg'`

Suppose we were to integrate both sides:

`int d/(dx)(fg)dx=int gf'+int fg'`

The left hand side is fg by the definition of an antiderivative. Then subtracting we get

`int fg'=fg-int gf'`

Let f=x, df=dx and g=-cosx

`int (xsinx)dx=x(-cosx)-int (-cosx)dx`

`=-xcosx + sinx +C` where C is the constant of integration.

The difficult part is deciding on, or finding, the appropriate functions f and g. Note that letting f=sinx, df=cosx dx, g=x, dg=dx will not work, as the resulting integral is as hard or harder than the original. One basic tip is to choose f such that f' is simpler than f.

Another hint, though not always applicable, is to choose f such that it comes before g in this list of functions: logarithmic, inverse trigonometric, algebraic, trigonometric, exponential. In this problem, we choose f to be algebraic and g to be trigonometric.

Integrating by parts will not always work, and occasionally you might have to do multiple applications of integration by parts to get to an integral you can do.

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial Team