Integral Calculus

The study of integrals and their properties is known as integral calculus. It is primarily useful for:

• To compute f from f’ (i.e. from its derivative). If a function f is differentiable in the range under consideration, then f’ is specified in that range.
• To determine the region under a curve.

Integration

Integration is exactly the opposite of differentiation. Differentiation is the partition of a portion into a number of smaller parts, and integration is gathering tiny parts to create a whole. It is frequently applied to area calculations.

Definite Integral

A definite integral has a specified boundary beyond which the equation must be computed. The lower and upper limits of a function’s independent variable are defined, and its integration is represented using definite integrals.

Indefinite Integral

An infinite integral lacks a fixed boundary, i.e. there is no upper and lower limit. As a result, the integration value is always followed by a constant value. 

More articles on Integral Calculus, to get better understanding:

• Tangents and Normal
• Equation of Tangents and Normal
• Absolute Minima and Maxima
• Relative Minima and Maxima
• Concave Function
• Inflection Points
• Curve Sketching
• Approximations & Maxima and Minima – Application of Derivatives
• Integrals
• Integration by Substitution
• Integration by Partial Fractions
• Integration by Parts
• Integration using Trigonometric Identities
• Functions defined by Integrals
• Indefinite Integrals
• Definite integrals
• Computing Definite Integrals
• Fundamental Theorem of Calculus
• Finding Derivative with Fundamental Theorem of Calculus
• Evaluation of Definite Integrals
• Properties of Definite Integrals
• Definite Integrals of Piecewise Functions
• Improper Integrals
• Riemann Sum
• Riemann Sums in Summation Notation
• Definite Integral as the Limit of a Riemann Sum
• Trapezoidal Rule
• Areas under Simple Curves
• Area Between Two curves
• Area between Polar Curves
• Area as Definite Integral
• Basic Concepts of differential equations
• Order of differential equation
• Formation of a Differential Equation whose General Solution is given
• Homogeneous Differential Equations
• Separable Differential Equations
• Linear Differential Equations
• Exact Equations and Integrating Factors
• Particular Solutions to Differential Equations
• Integration by U-substitution
• Reverse Chain Rule
• Partial Fraction Expansion
• Trigonometric Substitution
• Implicit Differentiation
• Implicit Differentiation – Advanced Examples
• Disguised Derivatives – Advanced differentiation
• Differentiation of Inverse Trigonometric Functions
• Logarithmic Differentiation
• Antiderivatives

Calculus Formulas

The Calculus formulas used in calculus can be divided into six major categories. The six major formula categories are limits, differentiation, integration, definite integrals, application of differentiation, and differential equations. 

Limits Formulas

Limits Formulas help in estimating the values to a definite number and are defined either to zero or to infinity. 

• Lt_x⇢0(xⁿ – aⁿ)(x-a)=na^(n-1)
• Lt_x⇢0(sin x)/x = 1 
• Lt_x⇢0(tan x)/x = 1
• Lt_x⇢0(e^x – 1)/x = 1
• Lt_x⇢0(a^x – 1)/x = log_ea
• Lt_x⇢0(1 +(1/x))^x = e
• Lt_x⇢0(1 + x)^1/x = e
• Lt_x⇢0(1 + (a/x))^x= e^a

Differentiation Formulas

Differentiation Formulas can be applied to algebraic expressions, trigonometric ratios, inverse trigonometry, and exponential terms.

Integration Formula

Integration Formulas can be derived from differentiation formulas, and are complimentary to differentiation formulas.

• ∫ xⁿ.dx = x^{n + 1}/(n + 1) + C
• ∫ 1.dx = x + C
• ∫ e^x.dx = e^x + C
• ∫(1/x).dx = log|x| + C
• ∫ a^x.dx = (a^x/log a) + C
• ∫ cos x.dx = sin x + C
• ∫ sin x.dx = -cos x + C
• ∫ sec²x.dx = tan x + C
• ∫ cosec²x.dx = -cot x + C
• ∫ sec x.tan x.dx = sec x + C
• ∫ cosec x.cotx.dx = -cosec x + C

Definite Integrals Formulas

Definite Integrals are the basic integral formulas with limits. There is an upper and lower limit, and definite integrals, that are helpful in finding the area within these limits.

Fundamental theorem of calculus =

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Differential Equations formula

Differential equations can be compared to general equations because they are higher-order derivatives.

In the general equation, the variable ‘x’ is an unknown, and in this case, the variable is the differentiation of dy/dx.

1. Homogeneous Differential Equation : f(λx, λy)= λⁿf(x,y)
2. Linear Differential Equation: dy/dx +Py = Q
3. The general solution of the Linear Differential Equation is y.e^-∫P.dx = ∫(Q.e^∫P.dx ).dx + C

In mathematics, Calculus deals with continuous change. It is also called infinitesimal calculus or “the calculus of infinitesimals”. The Two major concepts of calculus are Derivatives and Integrals.

• The derivative gives us the rate of change of a function. It describes the function at a particular point while the integral gives us the area under the curve.
• The integral gives us the area under the curve. Integral gathers the different values of a function over a number of values.

In general, classical calculus Calculus is the study of the continuous change of functions. In this article, we have provided everything related to Math Calculus for Beginners. Definition, examples, and practice questions will help you not only learn calculus theory but also practice calculus.