Fundamental Theorem of Calculus (Parts 1 and 2) Explained
The Fundamental Theorem of Calculus links derivatives and integrals. Both parts explained with intuition, formulas and worked examples.
Category: Integrals
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Improper Integrals: How to Tell If They Converge or Diverge
Improper integrals have infinite limits or infinite integrands. Learn the limit definition, the p-integral test, comparison, and 6 worked examples.
Partial Fraction Decomposition: Steps and Examples
How to integrate rational functions with partial fractions: distinct linear factors, repeated factors and irreducible quadratics, with worked examples.
Integration by Parts: Formula, LIATE Rule and Examples
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U-substitution reverses the chain rule. A 5-step method for choosing u, 7 worked examples, definite integrals with changed limits, and common mistakes.
Integral of √(1−x²): Trig Substitution and the Semicircle
∫√(1−x²)dx = ½(x√(1−x²) + arcsin x) + C. Full trig substitution, the semicircle area π/2 shortcut, and the general √(a²−x²) formula.
Integral of sec²(x) = tan(x) + C, with Examples
The integral of sec²(x) is tan(x) + C, because the derivative of tan x is sec²x. Examples with sec²(3x), tan²x, and definite integrals.
Integral of x·e^x: Integration by Parts Step by Step
∫x·e^x dx = x·e^x − e^x + C. A clear integration by parts walkthrough with LIATE, x²e^x, x·e^(2x), x·e^(−x) and a definite integral.
Integral of e^(−x²): The Gaussian Integral = √π
e^(−x²) has no elementary antiderivative, but its integral over the whole real line equals √π. See the polar-coordinates proof and the erf function.
Integral of 1/x: Why It’s ln|x| + C (Not x⁰/0)
The integral of 1/x is ln|x| + C. Why the power rule fails at n = −1, why the absolute value matters, and examples like 1/(3x+2).