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.

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:
- egin{aligned} ext{Let } u &= f(x), \ v &= g(x). \ ext{Then, the equation becomes:} \ rac{d}{dx}[uv] &= u rac{dv}{dx} + v rac{du}{dx}. \ ext{This can be rearranged as:} \ uv &= rac{du}{dx}
  ightarrow u dv = d(uv) - v du. \ ext{Thus, the formula for integration by parts is:} \ ext{Formula 1: } \ \ ext{Integrate:} & \
  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 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:
- egin{aligned} \int x ext{ sin}(x) \, dx &= -x ext{ cos}(x) - \int - ext{ cos}(x) \, dx \ &= -x ext{ cos}(x) + ext{sin}(x) + C, \ ext{ where } C & 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.

When evaluating definite integrals, the integration by parts formula can be adapted:
- egin{aligned} \ \ ext{If } u &= f(x) , \ v = g(x), \ ext{then we also have:} \ \inta^b u \, dv = uv |{a}^{b} - \int_a^b v \, du. \ \end{aligned}

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:
- egin{aligned} \ \int an^{-1}(x) \, dx &= 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
  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.

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.

Choose Functions: Let:
- u = ext{sin}(n-1)(x), \ dv = ext{sin}(x) \, dx.
Find Derivatives:
- du = (n - 1) ext{sin}(n-2)(x) ext{ cos}(x) dx,
- v = - ext{cos}(x).
Using Integration by Parts Gives:
- egin{aligned} \ \int ext{sin}^n(x) dx &= - ext{cos}(x) ext{sin}^{n-1}(x) + \int (n - 1) \text{sin}^{n-2}(x) \text{cos}^2(x) \, dx. \end{aligned}
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.