silemus.
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  1. 4L'Hopital's rule, limit
  2. 6Limit of (1−cos⁡x)/x2(1 - \cos x)/x^2
  3. 16Implicit differentiation, folium
  4. 21Maximizing rectangle area
  5. 22Optimizing box volume
  6. 23Related rates, sphere volume
  7. 24Related rates, circle area
  8. 27Integral of ln⁡x\ln x by parts
  9. 28Integration by parts
  10. 30Integral of sin⁡2x\sin^2 x
  11. 35Area between x2x^2 and xx
  12. 37Volume by disk method
  13. 41Telescoping series sum
  14. 44Maclaurin series of cos⁡x\cos x
  15. 45Radius of convergence, ratio test
  16. 49Double integral, inner yy first
  17. 50Separable ODE dy/dx=2ydy/dx = 2y

28Integration by parts

Use integration by parts to evaluate ∫01xex dx\displaystyle \int_0^1 x e^x\,dx.

Split x·eˣ using parts: u = x, dv = eˣdx, so du = dx, v = eˣ. This gives [xeˣ]₀¹ − ∫eˣdx = e − (e−1) = 1. Checking: e − (e−1) ≈ 2.718 − 1.718 = 1.

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