Integrals
Antiderivatives, integration techniques, the fundamental theorem and areas between curves.
Problems
- Integration by Parts with f·f′
- Antiderivative of Reciprocal Sine Addition
- Two Integrals That Nearly Cancel
- e^x sin²x on [0, π]
- Substitution then Parts
- Parts with an Integral Function
- Arcsine Plus Its Partner
- x⁴√(1 − x²) on [0, 1]
- An Even Part and an Odd Part
- Integrating by Parts Twice
- A Riemann Sum for x ln x
- Using f′(0) > 0
- x²√(4x − x²) on [0, 4]
- Making an Integral Function Increasing
- Recovering f from a Double Integral
- Derivative of an Inverse with an Integral
- Differentiating an Integral with Variable Limit
- Differentiating an Integral Identity
- Derivative from an Integral Equation
- Sum from an Integral Identity
- An Integral Identity with Unknown Constants
- Inverse Derivative from an Integral Equation
- Constants Hidden in an Integral Identity
- A Convolution Integral
- A Nested Integral's Derivative
- Derivative of an Inverse at Zero
- A Riemann Sum with a Parameter
- Σ i/(n² + i²)
- Riemann Sum of a Rational Expression
- Riemann Sum with a Squared Factor
- Riemann Sum Giving an Arctangent
- Σ n/(n² + k²)
- A Riemann Sum for x ln(1 + x)
- Slicing a Trapezoid
- Distances to a Curve
- Triangles on a Semicircle
- Even, Odd and Three Riemann Sums
- Volume by Shells about x = 5
- Area Between Two Parabolas
- Area of a Looped Curve
- Area Between a Cubic and Its Tangent
- Area Between a Hyperbola and Two Rays
- Area Between a Parabola and a Witch
- Area Between Two Root Curves
- A Solid with Square Cross-Sections
- Filling a Container
- Area Enclosed by Four Curves
- Area Cut by a Tangent from the Origin
- A Tangent, a Curve and the y-Axis
- Difference of Two Definite Integrals
- Solving x f(x) through an Integral
- Integral of a Periodic Function
- Scaling an Integral of f(x)/x
- Undoing Derivatives and Integrals
- Cube Root of a Shifted Cubic
- Area Under a Quarter Circle
- An Odd Integrand
- Integral of a Maximum
- A Substitution with cos³
- Completing the Square under a Root
- An Odd Part and a Semicircle
- |x² − 2x| on [−1, 2]
- An Odd Part Vanishes
- Sum over Sine-Power Antiderivatives
- Integral of a Cotangent Ratio
- Integrals over Unit Intervals
- Integrals of a Function and Its Inverse
- Integral of an Absolute Value
- A Reciprocal Pair of Log Integrals
- A Symmetric Integral with ln 6
- Integral with sin x + cos x over sin 2x
- An Integral Equation with an Exponential Kernel
- x over 1 + sin x on [0, π]
- Composing a Logistic Integral
- sin 5x over sin x on [0, 2π]
- Integrals of a Shifted Function
- Nested Absolute Values
- A Separable Integral Equation
- Sum of Tangent-Power Integrals
- An Identity with ∫ t f′(t) dt
- A Function Symmetric About x = a
- A Logarithmic Arcsine Integral
- 1/(2 + cos 2x) over [0, π]
- A Series from Tangent Integrals
- x⁷e^(−x²) on [0, ∞)
- A Function Containing Its Own Integral
- A Symmetry Worth Ten
- Areas Under a Fading Cosine
- Integral Involving an Inverse Function
- Integral from Two Functional Equations
- An Integral Equation with a Sine Kernel
- Integral of a Log of Root Sums
- An Integral Identity with a Scaled Argument
- The King Property with x sin x
- An Integral of the Inverse of eˣ + x − 1
- A Dirichlet-Type Integral
- Integrating g′ of an Inverse Tangent
- An Integral Equation with Two Unknown Integrals
- An Exact Derivative in Disguise
- An Arcsine Hidden in a Trig Integral
- An Integral Built from an Inverse
- A System of Product-Rule Equations
- A Derivative Known Only Up to Sign
- A Classic: ln(1 + tan x)
- A Shifted Polynomial Glued Together
- Equal Areas of |f′|
- A Product of Two Integrals
- A Cubic Symmetric About a Point
- Local Maxima of an Integral Function
- An Integral of an Absolute Difference
- A Function Hidden in an Integral Equation
- Three Conditions Fixing a Function
- A Cylinder Cut at 60°
- A Line Cutting a Wave Evenly
- Equilateral Slices
- An Integral That Measures Length
- A Function Defined by Its Own Integral
- A Quadratic Read Off Two Integrals