Symbolic Verification d/dx [x·arctan(x) − ½ln(1 + x²)] = arctan(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).
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)
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)
Rapidly chirping optical diffraction integral testing midpoint/trapezoid cancellation errors against 192-node Gauss–Legendre quadrature.
sin(x^2)