A Complete Exploration of Integration by Parts
Integration by parts is a mathematical technique used to integrate products of functions. It’s based on the product rule for differentiation and is often used when an integral cannot be easily solved by basic methods. The formula for integration by parts is derived from the differentiation of a product of two functions.
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Integration by Parts Formula:
The formula for integration by parts is derived from the product rule of differentiation, which states that \((uv)’ = u’v + uv’\). Integrating both sides, we get \(\int (uv)’ \, dx = \int u’v \, dx + \int uv’ \, dx\). This simplifies to \(uv = \int u’v \, dx + \int uv’ \, dx\). Rearranging, we get the integration by parts formula:
\(\int u \, dv = uv – \int v \, du\)
Here, \(u \) and \( dv \) are functions of \(x \). The derivatives \( du \) and \( v \) are \( du = u'(x) \, dx \) and \( v = \int dv \).
Choosing \( u \) and \( dv \):
The choice of \(u\) and \( dv \) is crucial. A common mnemonic for choosing \( u \) is “LIATE” (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential), which prioritizes functions to differentiate in that order. The idea is to choose \( u \) to be a function that becomes simpler when differentiated.
Example:
Let’s integrate \(\int x \, e^x \, dx\).
- Choose \( u \) and \( dv \):
- Let \( u = x \). Then, \(du = dx\).
- Let \( dv = e^x \, dx \). Then, \( v = \int e^x \, dx = e^x \).
- Apply the formula:
- Using the integration by parts formula, we have:
- \(\int x \, e^x \, dx = x \cdot e^x – \int e^x \cdot dx\)
- Solve the remaining integral:
- The integral of \( e^x \) is \( e^x \), so:
- \(\int x \, e^x \, dx = x \cdot e^x – e^x + C\)
Here, \( C \) is the constant of integration. This result gives the integral of the product of \( x \) and \( e^x \).
Integration by parts is a powerful tool in calculus, especially when dealing with products of different types of functions. The key to its successful application lies in the judicious choice of \( u \) and \(dv\).
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