Mastering Integration By Parts: A Comprehensive Guide To Calculus Integration

Integration by parts formula

∫udv= uv-∫vdu

Integration by parts is a technique used in calculus to find the integrated form of the product of two functions. The integration by parts formula for indefinite integrals is as follows:

∫u dv = uv – ∫v du

where u and v are two functions that are chosen according to a specific order of priority, and du and dv are their respective differentials.

The formula can also be written as:

∫f(x) g'(x) dx = f(x) g(x) – ∫g(x) f'(x) dx

This formula can be used to integrate a wide range of functions, including those that cannot be integrated by other methods. It is typically used when one of the functions in the integrand is easy to differentiate, but difficult to integrate, and the other function is easy to integrate but difficult to differentiate.

To apply the integration by parts formula, the first step is to choose u and dv. In general, u should be chosen based on its position in the ILATE acronym, which stands for Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential. The function with higher priority should be chosen as u. Then, dv is the differential of the second function. Next, it’s necessary to find du and v by taking the derivative and integral of u and dv, respectively. Finally, the integration by parts formula is applied to the original equation.

It should be noted that this technique only works for indefinite integrals. For definite integrals, the formula can be modified to include limits of integration and evaluated using the Fundamental Theorem of Calculus.

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