f(x)=1,∀x∈(a, b).
R(f, τ ) = Pn
i=1 f(ξi)4xi=Pn
i=1 4xi
= (x1−x0)+(x2−x1) + ...(xn−1−xn−2)+(xn−xn−1)
=xn−x0=b−a7→ b−a, d 7→ 0.
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Integration by Parts: A Comprehensive Guide with Examples, Cheat Sheet of Differential and Integral Calculus
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Typology: Cheat Sheet
2022/2023
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f (x) = 1, ∀x ∈ (a, b).
R(f, τ ) = ∑ni=1 f (ξi) 4 xi = ∑ni=1 4 xi = (x 1 − x 0 ) + (x 2 − x 1 ) + ...(xn− 1 − xn− 2 ) + (xn − xn− 1 ) = xn − x 0 = b − a 7 → b − a, d 7 → 0.
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- Integration by Parts ∫ (^) b a^ f^ (x)g
′(x)dx = f (x)g(x)|ba −^ ∫^ b a^ f^
′(x)g(x)dx or (^) ∫ (^) b
a^ f^ (x)dg(x) =^ f^ (x)g(x)|
ba −^ ∫^ b Examples: Evaluate the integrals a^ g(x)df^ (x)
- ∫^01 xexdx Integrating by parts, we get ∫ (^01) xexdx = ∫ (^01) xdex (^) = xex| (^10) − ∫ (^01) exdx = (1e^1 − 0 e^0 ) − ex|^10 = e − (e^1 − e^0 ) = e − e + 1 = 1.
- ∫^0 π x cos xdx Integrating by parts, we get ∫ (^0) π x cos xdx = ∫ (^0) π xd sin x = x sin x|π 0 − ∫ (^0) π sin xdx = (π sin π − 0 sin 0) + cos x|π 0 = cos π − cos 0 = − 1 − 1 = − 2.