Show carefully how the formula for "integration by parts" may be deduced from the Product Rule of differential calculus. Determine the following integrals: ∫ dx; ∫ x*e dx; ∫ e^x*sin(x/2) dx:
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In the standard notation, we write: $$\int u dv = uv - \int v du$$ Now, let's determine the given integrals: Show more…
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Key Concepts
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(a) Recall that the formula for integration by parts is obtained from the Product Rule. Use similar reasoning to obtain the following integration formula from the Quotient Rule. $$\int \frac{u}{v^{2}} d v=-\frac{u}{v}+\int \frac{1}{v} d u$$ (b) Use the formula in part (a) to evaluate $\int \frac{\ln x}{x^{2}} d x$.
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Integration by Parts and Present Value
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