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Integration by parts

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In calculus, integration by parts is a technique of integration applicable to integrands consisting of a product that cannot be rewritten as one or more easily integrated terms — at least, not without difficulty. The technique is particularly useful in cases containing a product of algebraic and transcendental factors.

Given two differentiable functions u and v,

\int u\ dv = uv - \int v\ du.

To use the technique, one identifies suitable functions u and dv and then differentiates u to get du and integrates dv to get v — ignoring the usual constant of integration term ("+\,C"), since it does not affect the final answer.

Note that the rule can also be written

\int f(x)\,g'(x)\ dx = f(x)\,g(x) - \int f'(x)\,g(x)\ dx,

for differentiable functions f and g.

DerivationEdit

According to the product rule for differentiation, given two differentiable functions u and v,

{d \over dx} (uv) = u{dv\over dx}+v{du\over dx} (alternatively, (uv)' = uv' + vu').

Therefore,

\int {d \over dx} (uv)\ dx = \int u{dv\over dx}\ dx + \int v{du\over dx}\ dx,

for suitably chosen antiderivatives. Simplifying the right-hand side of the equation,

\int {d \over dx} (uv)\ dx = \int u\ dv + \int v\ du.

On the left-hand side, the integral clearly "undoes" the differentiation (by the fundamental theorem of calculus), so

uv + C = \int u\ dv + \int v\ du.

Because the two antiderivative terms can always be chosen to make C=0, this can be simplified to:

uv = \int u\ dv + \int v\ du.

Solving for \int u\ dv, one obtains the final form of the rule:

\int u\ dv = uv - \int v\ du.

ExamplesEdit

Polynomial factor to a large powerEdit

A fairly simple example of integration by parts is the integral

\int x(x+3)^7\ dx.

Although the integrand only involves algebraic functions, it is a good candidate for the method because expansion of (x+3)^7 would be very tedious.

The key to the successful use of integration by parts is finding a usable value for dv. Doing so is something of an art and may require trial and error.

First consider a wrong way to do this integral by parts:

  • Let dv = x\ dx and u = (x+3)^7 (since it is the term left over after dv is determined).
  • Thus v = \frac{x^2}{2} and du = 7(x+3)^6\ dx.

Then

\int u\ dv = (x+3)^7\left(\frac{x^2}{2}\right) - \int \left(\frac{x^2}{2}\right) \cdot 7(x+3)^6\ dx.

However, it would be difficult to integrate the second term of the right-hand side of the equation, so this approach will be abandoned.

Here is a better way to handle this case:

  • Let dv = (x+3)^7 dx and u = x.
  • Thus v = \frac{(x+3)^8}{8} and du = dx.

Then

\int u\ dv = (x)\left[\frac{(x+3)^8}{8}\right] - \int \frac{(x+3)^8}{8}\ dx
{} = x\frac{(x+3)^8}{8} - \frac{(x+3)^9}{72} + C.

Here integration by parts works quite nicely. It can be easily confirmed by differentiation that the resulting antiderivative is correct.

Note that this integral may also be evaluated using the simpler integration by substitution technique.

Algebraic and transcendental factorsEdit

As another example where integration by parts is useful (and, in fact, necessary), consider the integral

\int x^2\sin x\ dx.

Choosing dv = x^2\ dx fails, as in the previous (counter)example, since the resulting integral is more difficult than the original. Instead:

  • Let dv = \sin x\ dx and u = x^2.
  • Thus v = -\cos x and du = 2x\ dx.

Then

\int x^2\sin x\ dx = \int u\ dv
{} = (x^2)(-\cos x) - \int -\cos x 2x\ dx
{} = -x^2 \cos x + \int 2x \cos x\ dx.

In this case, the second term in the final expression requires another application of integration by parts:

  • Let dv = \cos x\ dx and u = 2x.
  • Thus v = \sin x and du = 2dx.

Then

\int 2x \cos x\ dx = \int u\ dv
{} = (2x)(\sin x) - \int \sin x \cdot 2\ dx
{} = 2x\sin x + 2\cos x + C.

Substituting the last expression into the previous result:

\int x^2\sin x\ dx = -x^2\cos x + 2x\sin x + 2\cos x + C.

Note that if the second integration by parts step had instead used dv = 2x\ dx and u = \cos x, this would have "undone" the first step and we would have ended up with an integrand very much like the one we started with:

\int 2x \cos x\ dx = \int u\ dv
{} = (\cos x)(x^2) - \int x^2 (-\sin x)\ dx
{} = 2x\sin x + 2\cos x + C.

Thus, the correct choice of u and dv is particularly important when multiple applications of the technique are required. In general, if u is chosen to be an algebraic function in the first step, it should be algebraic in all subsequent steps.

Choosing u and dvEdit

Fortunately, there is a mnemonic for choosing u and dv, which covers a large variety of integrands:

uL I A T Edv

The letters stand for:

This mnemonic only works when the integrand is the product of two different types of factors. The factor whose type of function appears higher in this list should generally be chosen as u, the factor whose type appears lower as dv.

For example, in the integral

\int x^2\ln x\ dx,

the choices should be u=\ln x, since this is a logarithmic function, and dv=x^2\ dx, since this is an algebraic one ("L" appears before "A" in the mnemonic). On the other hand, in the integral

\int x^2e^{-x}\ dx

the proper choices are u=x^2 (algebraic) and dv=e^{-x}\ dx (exponential).

Note that the second example above also follows the rule suggested by this mnemonic.

If the mnemonic doesn't seem to work for a given integral it is possible that it may be a simpler form that can be evaluated using the substitution method, or perhaps rewritten into a simpler form using algebraic or trigonometric techniques (e.g., trigonometric identities).

A slightly different mnemonic that works almost as well — and has the added benefit of sounding more like an English word — is:

uL I P E Tdv

Here the "P" stands for Power, which includes polynomials and roots (fractional powers). The other letters are as above.

Notice that the last two letters are switched in this form; this is usually not an issue, since integrals involving a product of trigonometric and exponential factors can generally be done "either way" (with respect to the choice of u and dv) or not at all using this technique.

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