This book is designed as an advanced comprehensive guide of integral calculus. This covers the following topics: indefinite or antiderivative integral, basic indefinite integrals, properties, integration methods: decomposition, change of variable or substitution, integration by parts, trigonometric, rational functions and partial fractions, substitutions to rationalize, definite integral, area by mean of inscribed and circumscribed polygons, Riemann sums, integrability theorem, properties, first fundamental theorem, second fundamental theorem or Barrow's rule, integral as area under the curve, area between two curves, mean value theorem for integrals, theorem of symmetry and periodicity, volumes and volumes of revolution: discs, washers, cylindrical shells, arc length and lateral surface of revolution, indeterminate forms of type 0|0 and ∞|∞, improper integrals: infinite limits of integration, infinite integrands, integrands with functions that are infinite at an interior point, derivatives and integral of transcendental functions: logarithmic, exponential, trigonometric, hyperbolic, inverse, introduction to differential equations, miscellaneous exercises. This is including the deduction of many formulas to clarify the concepts and meet all the needs of the students to reach the fundamentals of integral calculus. Finally, ther objective of the book is to reach the interest and motivation of the students by the real nature of the integral calculus.