Integration by parts formula
Integration by parts is a method used to integrate a product of two functions
Integration by parts is a method used to integrate a product of two functions. The formula for integration by parts is as follows:
∫ u dv = uv – ∫ v du
Here, u and v are functions of the variable of integration, and du and dv represent their differentials.
To use the integration by parts formula, you need to identify which function to assign as u and which one to assign as dv. Generally, the preference is to let u be the function that becomes simpler when differentiated, and dv be the function that becomes simpler when integrated.
Once you have identified u and dv, you can calculate du and v by differentiating and integrating, respectively:
du = u’ dx (where u’ represents the derivative of u with respect to x)
v = ∫ dv (integrating dv)
Next, you substitute these expressions for du and v in the integration by parts formula:
∫ u dv = uv – ∫ v du
Finally, you can use the formula to evaluate the integral on the right-hand side. Sometimes, this process may require applying integration by parts again, depending on the complexity of the original integral.
It’s worth noting that sometimes the integration by parts formula needs to be applied multiple times, or additional algebraic manipulation might be necessary to simplify the integral before using the formula. Also, specific cases may require the use of tabular integration by parts, which is a common extension of the basic formula.
Overall, integration by parts is a powerful technique that allows you to compute integrals by breaking them down into simpler steps and applying differentiation and integration principles.
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