Integration by Parts | Explained with Examples

Integration by parts formula

The integration by parts formula is a technique in calculus that allows us to integrate the product of two functions

The integration by parts formula is a technique in calculus that allows us to integrate the product of two functions. It is derived from the product rule for differentiation.

The formula states:
∫u dv = uv – ∫v du

Here, u and v are two functions, and du and dv represent the differentials of u and v, respectively. The symbol ∫ represents the integral operator.

To use the integration by parts formula, you need to choose which function to differentiate (u) and which function to integrate (dv). This choice is usually based on a specific strategy called the “ILATE” rule, which stands for inverse trigonometric functions, logarithmic functions, algebraic functions, trigonometric functions, and exponential functions. The idea is to choose u in such a way that when differentiated, it becomes simpler or is reduced to zero.

Once you have identified u and dv, you differentiate u and integrate dv separately. Then substitute these values into the formula to find the integral of the product.

Let’s go through an example to illustrate how the integration by parts formula works:

Example: Find ∫x cos(x) dx

Solution:
In this case, we can choose u = x and dv = cos(x) dx. Therefore, du = dx and v = ∫cos(x) dx = sin(x).

Using the integration by parts formula, we have:
∫x cos(x) dx = uv – ∫v du
= x sin(x) – ∫sin(x) dx

The integral on the right side, ∫sin(x) dx, can be easily evaluated as -cos(x). Substituting this back into the equation, we have:
∫x cos(x) dx = x sin(x) + cos(x) + C

Where C is the constant of integration.

So, the integral of x cos(x) dx is x sin(x) + cos(x) + C.

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