EE102 Systems and Signals (Discussion)

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Integration by Parts

The product rule of differentiation is

$$\frac{d(uv)}{dt} = v \frac{du}{dt} + u \frac{dv}{dt}$$

Integrating both sides and rearranging terms, we get the formula for integration by parts:

$$\int u \frac{dv}{dt} dt = \int \frac{d(uv)}{dt} dt - \int v \frac{du}{dt} dt$$

$$= uv - \int v du$$

I like the following form better:

$$\int f(t) g(t) dt = f(t) \int g(t) dt - \int f'(t) \left(\int g(t) dt\right) dt$$

To apply integration by parts to integral of a product of two functions $f(t)g(t)$, first $\int g(t) dt$ should be easy to find, and second, the second integral on the RHS should be easier to compute that the original integral.

So, when you have a product of two functions, $g(t)$ is usually chosen to be the exponential or sinusoidal function (if it is one of the functions in the product) and $f(t)$ is usually a function whose derivative is simpler to integrate.

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