Indefinite Integrals

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Properties of indefinite integral

(1) Let f(x) be a real value differentiable function, then

Proof:

Let F(x) be any anti derivative of f(x)

Similarly we know the

Where C is constant of integration.

(2) Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent.

Proof:

Let the two indefinite integrals be

Given:

Where C is any number.

or

The family of curves

are identical.

(3)

Proof:

By property (1), we have

Also we have

= f(x) + g(x) …(2)

From (1) and (2), we have

(4) For any real number k,

By property (1)

Also

= kf(x) …(ii)

From (i) and (ii)

[Note that while using property (2), we can express two equivalent integrals by writing without mentioning the constants)

More generally, combining property (2) and property (3), we can write

Where f₁, f₂, ….f_n are functions and k₁, k₂, …k_n are real numbers.

Example: Write an anti derivative of sin2x - 4e^3x using method of inspection.

The anti derivative of sin2x is

Similarly

Comparison between differentiation and integration

1. Both are operations on functions.

2. Both are linear. This is because of the following:

(i)

(ii) The constant can be taken outside the differential as well as integral sign as shown below:

3. We heve already seen that not all functions are differentiable. Similarly, all functions are not integrable. We will learn about non-differentiable and non-integrable functions in our higher classes.

4. The derivative of a function, when it exists is a unique function. The integral of a function is not so. However, it always differs by a constant only.

5. When a polynomial function P is differentiated, the result is a polynomial whose degree is 1 less than the degree of P. When a polynomial function P is integrated, the result is a polynomial whose degree is 1 more than that of P.

6. We can speak of the derivative of a function at a point. We never speak of the integral of a function at a point, we speak of the integral of a function over an interval on which integral is defined.

7. The derivative of a function has a geometrical meaning, namely, the slope of the tangent to the corresponding curve at the point. Similarly, indefinite integral of a function represents geometrically, a family of curves placed parallel to each other having parallel tangents at the points of intersection of the curves of the family with the lines orthogonal to the axis representing the variable of integration.

8. The derivative is used for finding some physical quantities like the velocity of a moving particle, when the distance traversed at any time t is known. Integral is used to find the distance travelled on time t when velocity at time t is known n.

9. Differentiation is a process involving limits, so is integration.

10. The process of differentiation and integration are inverses of each other.

In the earlier section we have found the integral (antiderivative) of a function by inspection. For a given function f, it may be difficult to find F such that by inspection. Therefore we need to learn different methods of integration in this section. The three different methods of integration, we learn are

1. Method of substitution

2. Integration using partial fraction.

3. Integration by parts.