Full-Wave Bridge Rectifier |sin x| Fourier Spectrum — DC Component 2/π & Ripple
Extracts DC average a₀/2 = 2/π ≈ 0.63662 plus even ripple harmonics cos(2nx)/(4n² − 1) for AC-DC filter capacitor design.
Pre-Computed Analytical Result
More Calculus & Series Workbenches
Applies symbolic Product and Chain rules step-by-step, locating stationary extrema at x = 0 and x = 4 plus concavity inflections.
x^2 * exp(-0.5*x)
Derives exact rational first and second derivatives, curvature κ(x₀), and bounded global extrema without pole singularities.
(x^2 + 1) / (x^2 - x + 2)
Finds all stationary maxima, minima, roots, and concavity sign-flips across [−π, π] with live tangent line visualization.
sin(2*x) + 0.5*cos(x)
Classic integration-by-parts identity where terms x/(1+x²) cancel algebraically to leave f′(x) = arctan(x) and f″(x) = 1/(1+x²).
x*atan(x) - 0.5*ln(1 + x^2)
Evaluates ∫₋₂² e^(−x²) dx = √π·erf(2) ≈ 1.76416278 across 6 quadrature rules, revealing exponential accuracy gain.
exp(-x^2)
Shades and integrates the famous ±1.96σ two-tailed statistical threshold to 7-digit Gauss–Legendre precision.
exp(-x^2 / 2) / sqrt(2*pi)