# 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.

## 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.