The art of mastering the integration by parts

The integration parts allows to calculate the integral of the product of some functions. Let's derive a formula for it:

The formula for the derivative of a product is (uv)' = u'v + uv'

Then uv' = (uv)'-u'v

Let's integrate the 2 sides;

∫uv'dx = ∫(uv)'dx-∫u'vdx

∫u.(dv/dx).dx = uv-∫(du/dx).vdx

Let's simplify:

∫udv = uv-∫vdu

Example 1. Evaluate ∫xsinxdx;

Let's apply the formula: ∫udv = uv-∫vdu

Let's choose u = x and dv = sinxdx. Then du = dx  v = -cosx

Let's substitute u, v and dv in the formula:

∫xsinxdx = (x)(-cosx)- ∫(-cosxdx)dx
              = -xcosx + ∫cosxdx
              =  -xcosx + sinxdx + C

Note. Using the integration by parts can be tricky in choosing u and dv. To avoid bumping into a new integral that's not easy to calculate, choose u to have a derivative easier to calculate. Choose dv into an integral easy to calculate. In general you can use the following guide:
1) Choose u to be the part whose derivative is simpler than u. Use dv as the remaining term.
2) Choose dv to be the portion whose integral can be calculated using the basic formula. Choose u as the remaining term.

Application:  In example 1, choose u and dv differently to see if the integral is easy to calculate.

Example 2. Evaluate ∫xe^xdx ( read x exponential x dx). The sign ^ is used to signify that x is the exponent of e.

We can choose u first and choose dv as the remaining term. If so the derivative of u should be simpler than u. We can choose dv first so that the derivative can be calculated using a basic formula and choose u as the remaining portion of the integral.

Let's choose u = x and dv = e^x dx. Then du = dx and v = e^x.
Applying the formula:  ∫xe^xdx = xe^x-∫ e^x.dx
                                                   = xe^x- e^x + C
Interested in learning more about integrals visit Center for Integral Development