Integration by Parts

Integration by Parts

Integration by parts integrates products of functions. It is derived directly from the product rule of differentiation and transforms a difficult integral into a simpler one.

The Fundamental Formula

Let $u$ and $v$ be two differentiable functions of $x$. From the product rule:

Integrating both sides with respect to $x$:

Rearranging gives the integration by parts formula:

Standard Form

Let $u = f(x)$ and $\frac{dv}{dx} = g(x)$. Then $\frac{du}{dx} = f'(x)$ and $v = \int g(x)\,dx$. Substituting:

In words: the integral of the product equals (first function) × (integral of the second function) minus the integral of [(derivative of the first function) × (integral of the second function)].

Choosing the First and Second Functions

The choice of first function ($f$) and second function ($g$) is crucial — a wrong choice can make the integral more complicated instead of simpler.

Important Remarks

Remark (i): Applicability

Integration by parts is not applicable to all products of functions. For example, $\int\sqrt{x}\sin x\,dx$ cannot be evaluated by this method because there is no function whose derivative is $\sqrt{x}\sin x$.

Remark (ii): The Constant of Integration

When finding the integral of the second function, we do not add a constant of integration — it cancels out in the final result.

Verification: Suppose we write $\int\cos x\,dx = \sin x + k$ (where $k$ is any constant). Then:

The $kx$ terms cancel, confirming that adding a constant is unnecessary.

Standard Results