Integration By Parts

7 Techniques of Integration

7.1 Integration by Parts

Introduction to Integration by Parts

• Every differentiation rule has a corresponding integration rule.

  • Example:

  • The Substitution Rule for integration corresponds to the Chain Rule for differentiation.

  • The rule that corresponds to the Product Rule for differentiation is called integration by parts.

Integration by Parts: Indefinite Integrals

• The integration by parts formula is derived from the Product Rule for differentiation:

  • If $f$ and $g$ are differentiable functions, then
    $rac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$

• Rearranging this gives:

  • $\begin{aligned} ext{Let } u &amp;= f(x), \ v &amp;= g(x). \ ext{Then, the equation becomes:} \ rac{d}{dx}[uv] &amp;= u rac{dv}{dx} + v rac{du}{dx}. \ ext{This can be rearranged as:} \ uv &amp;= rac{du}{dx} <br>ightarrow u dv = d(uv) - v du. \ ext{Thus, the formula for integration by parts is:} \ ext{Formula 1: } \ \ ext{Integrate:} &amp; \ <br>t R \ u dv = uv - \int v du. \ \ \text{This is called the formula for integration by parts.}\ ext{It may be more easily remembered in the following notation.} \ ext{Let } u = f(x) ext{ and } v = g(x). \ ext{Then the differentials are } du = f'(x)dx ext{ and } dv = g'(x)dx.\ ext{So, by the Substitution Rule, the formula for integration by parts becomes:} \ ext{Integrate: } \ \ u dv = uv - \int v du. \ \ \ \ \end{aligned}$

Example 1: Finding an Integral Using Integration by Parts

• Example Statement: Find $\int x ext{ sin}(x) \, dx$.

• From the formula, choose:

  • $u = x, \ dv = ext{sin}(x) \, dx$.

  • Now find $du$ and $v$:

  • $du = dx$

  • $v = - ext{cos}(x)$

• Using Formula 1:

  • $\begin{aligned} \int x ext{ sin}(x) \, dx &amp;= -x ext{ cos}(x) - \int - ext{ cos}(x) \, dx \ &amp;= -x ext{ cos}(x) + ext{sin}(x) + C, \ ext{ where } C &amp; ext{ is the constant of integration.} \ \ \end{aligned}$

• Verification: It's wise to check the answer by differentiating the result:

  • Differentiate: $rac{d}{dx}[-x ext{ cos}(x) + ext{sin}(x) + C] = x ext{ sin}(x)$, confirming the expected result.

Integration by Parts: Definite Integrals

• When evaluating definite integrals, the integration by parts formula can be adapted:

  • $\begin{aligned} \ \ ext{If } u &amp;= f(x) , \ v = g(x), \ ext{then we also have:} \ \int<em>a^b u \, dv = uv |</em>{a}^{b} - \int_a^b v \, du. \ \end{aligned}$

Example 5: Definite Integral Evaluation

• Goal: Calculate $\int an^{-1}(x) \, dx$.

• Choose the following for the integration:

  • $u = an^{-1}(x), \ du = rac{dx}{1 + x^2}, \ dv = dx, \ v = x$.

• Using the integration by parts formula gives:

  • $\begin{aligned} \ \int an^{-1}(x) \, dx &amp;= x an^{-1}(x) - \int \frac{x}{1 + x^2} \, dx. \ \end{aligned}$

• Evaluating the Integral: The above integral can be solved using substitution. Let:

  • $t = 1 + x^2$

  • Thus, $dt = 2x \, dx <br>ightarrow dx = rac{dt}{2x}$; change limits accordingly for definite integration.

• The final evaluation leads to simplified forms and checks against initial conditions to find definite integral solutions.

Reduction Formulas

• Integration by parts is often used to derive reduction formulas, allowing an integral to be expressed in terms of simpler integrals.

• Example Statement: Prove the reduction formula:
  $\int \sin^n(x) \, dx = \sin^{n-1}(x)(-\cos(x)) + (n-1)\int \sin^{n-2}(x) \, dx$,
  where $n \ ext{ is an integer } n \geq 2$.

Example 6: Proving the Reduction Formula

1. Choose Functions: Let:

   • $u = ext{sin}(n-1)(x), \ dv = ext{sin}(x) \, dx$.

2. Find Derivatives:

   • $du = (n - 1) ext{sin}(n-2)(x) ext{ cos}(x) dx$,

   • $v = - ext{cos}(x)$.

3. Using Integration by Parts Gives:

   • $\begin{aligned} \ \int ext{sin}^n(x) dx &amp;= - ext{cos}(x) ext{sin}^{n-1}(x) + \int (n - 1) \text{sin}^{n-2}(x) \text{cos}^2(x) \, dx. \end{aligned}$

4. Substituting and Rearranging leads to:

   • Deriving a final formula by explicitly solving the rearranged integral form per constants,

   • Showing dependence back on the original integral form.