Integrate using 'Integration by parts'

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.

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.
 * 4) Determine \(dU\).
 * 5) Evaluate \(\begin{aligned}\frac{dU}{dx}\end{aligned}\)
 * 6) \(\begin{aligned}\frac{dU}{dx}=\frac{d}{dx} x = 1 \end{aligned}\)
 * 7) Multiply each side by \(dx\) to give: \(dU = 1 \, dx\)
 * 8) Determine \(\begin{aligned}V = \int dV\end{aligned}\):
 * 9) 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.
 * 10) Insert them into the equation for integration by parts to get: \( \qquad \begin{aligned}\,\int x \sin(x) = U \, V - \int V \, dU = x (-\cos(x)) - \int (-\cos(x)) \, dx \qquad \end{aligned}\)
 * 11) Simplify.
 * 12) Cancel the signs and remove the brackets: \(\begin{aligned} \int x \sin(x) = - x \cos(x) + \int \cos(x) \, dx \end{aligned}\)
 * 13) 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.