Ohio Assessments for Educators (OAE) Mathematics Practice Exam

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What does the integration by parts formula state?

∫udv=uv+∫vdu
∫udv=uv-∫vdu
∫udv=uv
∫udv=∫u+∫v

The correct answer is: ∫udv=uv-∫vdu

The integration by parts formula is derived from the product rule of differentiation and is used to integrate the product of two functions. The correct statement for the integration by parts formula is that it expresses the integral of a product of functions $ u $ and $ dv $ as the product of the function $ u $ and its corresponding $ v $ (the integral of $ dv $), minus the integral of the product of $ v $ and the derivative of $ u $ (that is, $ du $). This leads to the formula: \[ \int u \, dv = uv - \int v \, du \] Here, $ u $ is a differentiable function and $ dv $ is an integrable function. The formula essentially shifts the integration from the original product to a new integral that is often easier to solve. The negative sign in the formula is crucial because it accounts for the way the product rule operates in differentiation. Therefore, this option captures the correct relationship needed to apply integration by parts successfully.