Integrate using 'Integration by parts'

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Understanding

  • Integration by parts is used to integrate a product of two continuously integrable and differentiable functions.
  • On this page, \(U(x)\) and \(V(x)\) are two differentiable functions.
  • According to the product rule of differentiation:

\( \begin{aligned} \frac{d}{dx}(U(x)V(x)) & = U(x)\frac{dV}{dx} + V(x)\frac{dU}{dx} \end{aligned} \)

where \( \begin{aligned} \frac{d}{dx}(U(x)V(x))\end{aligned} \) is the differential with respect to \(x\) of the product \(U(x)V(x)\).

If you integrate both sides you get:

\( \begin{aligned} U(x)V(x) & = \int U(x)\frac{dV}{dx}dx + \int V(x)\frac{dU}{dx}dx \end{aligned} \)

Rearranging to make the first term on the left the subject of the equation:

\( \begin{aligned} \int U(x)\frac{dV}{dx}dx & = U(x)V(x) - \int V(x)\frac{dU}{dx}dx \end{aligned} \)

The notation can be simplified to:

\( \begin{aligned} \int U \, dV & = U \, V - \int V \, dU \end{aligned} \)

This is the equation that's used to construct an integration by parts.

Method

Example: \( \begin{aligned}\int x \sin(x) \, dx \end{aligned} \)

  1. Write down the formula \( \begin{aligned} \int U \, dV = \, U \, V - \int V \, dU \end{aligned} \).
  2. Let \(\begin{aligned}U = x \end{aligned}\) and \(\begin{aligned}dV & = \sin(x) \, dx\end{aligned}\)
  3. Find the unknown terms of the integration by parts formula.
    1. Determine \(dU\).
      1. Evaluate \(\begin{aligned}\frac{dU}{dx}\end{aligned}\)
        1. \(\begin{aligned}\frac{dU}{dx}=\frac{d}{dx} x = 1 \end{aligned}\)
      2. Multiply each side by \(dx\) to give: \(dU = 1 \, dx\)
    2. Determine \(\begin{aligned}V = \int dV\end{aligned}\):
      1. Evaluate \(\begin{aligned}\int dV = \, \int \sin(x) \, dx = -\cos(x) + C\end{aligned}\) for an indefinite integral or \(-\cos(x)\) for a definite integral, where C is a constant.
  4. Insert them into the equation for integration by parts to get: \( \qquad \begin{aligned}\,\int x \sin(x) = U \, V - \int V \, dU = x (-\cos(x)) - \int (-\cos(x)) \, dx \qquad \end{aligned}\)
  5. Simplify.
    1. Cancel the signs and remove the brackets: \(\begin{aligned} \int x \sin(x) = - x \cos(x) + \int \cos(x) \, dx \end{aligned}\)
    2. Integrate \(\cos(x)\): \(\begin{aligned}\qquad \qquad \qquad \qquad \qquad \qquad \quad \quad = - x \cos(x) + \sin(x) + D \end{aligned}\) , where \(D\) is a constant.
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