- 27-Jun-2022

Central government consists of **all administrative departments of the state and other central agencies whose responsibilities cover the whole economic territory of a country, except for the administration of social security funds**.

Ministers of India The Government in India or the central or the union government is divided into three main sections namely the **executive, legislature and the judiciary** shown as under. The responsibility of each section of the government is also mentioned along.

Central government consists of **all administrative departments of the state and other central agencies whose responsibilities cover the whole economic territory of a country, except for the administration of social security funds**.

**The three spheres of Government**

- National Government.
- Provincial Government.
- Local Government.

An indefinite integral is **a function that takes the antiderivative of another function**. It is visually represented as an integral symbol, a function, and then a dx at the end. The indefinite integral is an easier way to symbolize taking the antiderivative.

The Integral command is one of the most significant commands on the TI-Nspire CAS Calculus submenu. **Press →Calculus→Integral** to open the Integral command.

To summarize, you could say that **an antiderivative is just one function having a given derivative, whereas an indefinite integral is the collection of all functions having a given derivative**.

An indefinite integral, sometimes called **an antiderivative**, of a function f(x), denoted byis a function the derivative of which is f(x).

Example: **F(x)=x3 is an antiderivative of f(x)=3x2**. Also, x3+7 is an anti-derivative of 3x2, since d(x3)dx=3x2 and d(x3+7)dx=3x2. The most general antiderivative of f is F(x)=x3+C, where c is an arbitrary constant.

**Substitution in the indefinite integral**

- Calculate the derivative of u, and then solve for "dx."
- Substitute the expression for u in the original integral, and also substitute for dx.
- Eliminate the variable x, if it is still present, leaving an integral in u only.
- Simplify the integrand.
- Evaluate the simplified integral.

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- The process of finding the indefinite integral is also called integration or integrating f(x). f ( x ) .
- The above definition says that if a function F is an antiderivative of f, then. ∫f(x)dx=F(x)+C. for some real constant C. C .
- Unlike the definite integral, the indefinite integral is a function.

An antiderivative is a function that **reverses what the derivative does**. One function has many antiderivatives, but they all take the form of a function plus an arbitrary constant. Antiderivatives are a key part of indefinite integrals.

**To find antiderivatives of basic functions, the following rules can be used:**

- x
^{n}dx = x^{n}^{+}^{1}+ c as long as n does not equal -1. This is essentially the power rule for derivatives in reverse. - cf (x)dx = c f (x)dx.
- (f (x) + g(x))dx = f (x)dx + g(x)dx.
- sin(x)dx = - cos(x) + c.

Antiderivatives, which are also referred to as indefinite integrals or primitive functions, is essentially the opposite of a derivative (hence the name). More formally, **an antiderivative F is a function whose derivative is equivalent to the original function f**, or stated more concisely: F′(x)=f(x).

**If we can find a function F derivative f, we call F an antiderivative of f**. for all x in the domain of f. Consider the function f(x)=2x. Knowing the power rule of differentiation, we conclude that F(x)=x2 is an antiderivative of f since F′(x)=2x.

Antiderivatives are related to **definite integrals** through the second fundamental theorem of calculus: the definite integral of a function over a closed interval where the function is Riemann integrable is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.

**An antiderivative is a function F(x) with the property that F′(x)=f(x)**. The Fundamental Theorem of Calculus tells us that we can compute definite integrals using antiderivatives, i.e. ∫baf(x)dx=F(b)−F(a).

Differentiation and Integration are the two major concepts of calculus. Differentiation is used to study the small change of a quantity with respect to unit change of another.

Differentiation and Integration Formulas.

Differentiation Formulas | Integration Formulas |
---|---|

d/dx cos x = -sin x | ∫ cos x dx = sin x + C |

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