Integration

• How to Integrate cos(1/x)
  The antiderivative of cos(1/x) is x cos(1/x) + Si(1/x) + C, where Si is the sine integral special function.
• How to Show That an Integral-Defined Function Is Strictly Decreasing
  For a function defined by an integral, prove strict decrease by comparing its integrands or justifying differentiation under the integral sign.
• How to Solve Integrals Using Series
  Use a power series to replace a difficult integrand with simpler terms, then integrate term by term after checking convergence on the interval.
• How to Prove That a Function Is Riemann Integrable
  Prove Riemann integrability by showing upper and lower sums can be made arbitrarily close, or use a standard sufficient condition.
• Evaluate the Integral of the Piecewise Function from −2 to 4
  Split the integral at the piecewise breakpoint, evaluate each part, and add the signed results to obtain −4/3.
• How to Compute ∫₀¹ xᵐ(1 − xⁿ)ᵖ dx
  For n > 0, m > −1, and p > −1, substitute t = xⁿ to evaluate the integral as a Beta function and, if desired, Gamma functions.
• How to Calculate an Ellipsoid’s Volume in Cylindrical Coordinates
  Set the ellipsoid's vertical bounds in cylindrical coordinates, include the Jacobian r, and integrate to obtain 4πabc/3 for semiaxes a, b, and c.
• What Is the Laplace Transform of $t e^{at}\sin(bt)$?
  The Laplace transform of t e^{at} sin(bt) is 2b(s−a)/([(s−a)^2+b^2]^2), for real a, nonzero b, and Re(s)>a.
• How to Set Up a Double Integral Over an Inequality-Defined Region
  To integrate over a region given by inequalities, sketch its boundary, find projection limits, and write the bounds in an order that covers it once.
• Reduction Formula for ∫ cos(nx)/cos(x) dx
  Derive the recurrence for the integral of cos(nx)/cos(x), with its base cases, a worked example, and the alternative cosine-power formula.