Properties of Indefinite Integral

1.Inverse of Differentiation

This shows that differentiation and indefinite integration are inverse processes.

2. Same Derivative Implies Same Family of Antiderivatives

If \[\frac{d}{dx}[F(x)] = \frac{d}{dx}[G(x)]\]

then \[F(x) = G(x) + C\]

So, two indefinite integrals of the same function differ only by a constant.

3. Sum Rule

This means integration is distributive over addition.

4. Difference Rule

This is the corresponding rule for subtraction.

5. Constant Multiple Rule

6. General Linearity Rule

where k and l are constants.

Find the following integrals:

1. \[\int (\sin x + \cos x) dx\]
2. \[\int \text{cosec } x (\text{cosec } x + \cot x) dx\]
3. \[\int \frac{1 - \sin x}{\cos^2 x} dx\]

Solution:

(i) We have

\[\int (\sin x + \cos x) dx = \int \sin x dx + \int \cos x dx\]

\[= -\cos x + \sin x + \text{C}\]

(ii) We have

\[\int \text{cosec } x (\text{cosec } x + \cot x) dx = \int \text{cosec}^2 x dx + \int \text{cosec } x \cot x dx\]

\[= -\cot x - \text{cosec } x + \text{C}\]

(iii) We have

\[\int \frac{1 - \sin x}{\cos^2 x} dx = \int \frac{1}{\cos^2 x} dx - \int \frac{\sin x}{\cos^2 x} dx\]

\[= \int \sec^2 x dx - \int \tan x \sec x dx\]

\[= \tan x - \sec x + \text{C}\]

Find the anti derivative F of \[f\] defined by \[f(x) = 4x^3 - 6\], where \[\text{F}(0) = 3\]

Solution: One anti derivative of \[f(x)\] is \[x^4 - 6x\] since

Therefore, the anti-derivative F is given by

\[\text{F}(x) = x^4 - 6x + \text{C}\], where C is constant.

Given that \[\text{F}(0) = 3\], which gives,

\[3 = 0 - 6 \times 0 + \text{C}\] or \[\text{C} = 3\]

Hence, the required anti-derivative is the unique function F defined by

\[\text{F}(x) = x^{4} - 6x + 3\].