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Integrals
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- Integration as an Inverse Process of Differentiation
- Properties of Indefinite Integral
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Properties of Indefinite Integrals
1.Inverse of Differentiation
This shows that differentiation and indefinite integration are inverse processes.
\[ \int f'(x)\, dx = f(x) + C \]
Thus, differentiation and integration are inverse processes of each other.
2. Same Derivative Implies Same Family of Antiderivatives
If \[\frac{d}{dx}[F(x)] = \frac{d}{dx}[G(x)]\]
then \[F(x) = G(x) + C\]
Thus, two indefinite integrals having the same derivative represent the same family of curves and are considered equivalent.
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
\[ \int [k_1 f_1(x) + k_2 f_2(x) + \cdots + k_n f_n(x)]\, dx = k_1 \int f_1(x)\, dx + k_2 \int f_2(x)\, dx + \cdots + k_n \int f_n(x)\, dx \]
where \[k_1, k_2, \ldots, k_n\] are real constants.
Integration by Method of Inspection
Method of Inspection:
To find an antiderivative of a function, we may look for a known function whose derivative is the given function. This is called integration by the method of inspection.
Example
Find an antiderivative of \[ \cos 2x. \]
Solution:
We know \[ \frac{d}{dx}(\sin 2x) = 2\cos 2x. \]
Therefore, \[ \int \cos 2x\, dx = \frac{1}{2}\sin 2x + C \]
Example 1
Find the following integrals:
- \[\int (\sin x + \cos x) dx\]
- \[\int \text{cosec } x (\text{cosec } x + \cot x) dx\]
- \[\int \frac{1 - \sin x}{\cos^2 x} dx\]
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
\[\int \frac{1 - \sin x}{\cos^2 x} dx = \int \frac{1}{\cos^2 x} dx - \int \frac{\sin x}{\cos^2 x} dx\]
\[= \int \sec^2 x dx - \int \tan x \sec x dx\]
\[= \tan x - \sec x + \text{C}\]
Example 2
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
\[\text{F}(x) = x^{4} - 6x + 3\].
Key Points: Properties of Indefinite Integrals
| Property | Formula |
| Reverse of differentiation | \[\frac{d}{dx}\left(\int f(x) dx\right) = f(x)\] |
| Same derivative | \[F'(x) = G'(x) \Rightarrow F(x) = G(x) + C\] |
| Sum rule | \[\int (f + g)dx = \int fdx + \int gdx\] |
| Difference rule | \[\int (f - g)dx = \int fdx - \int gdx\] |
| Constant multiple rule | \[\int k f(x)dx = k \int f(x)dx\] |
| General linearity | \[\int (kf \pm lg)dx = k \int fdx \pm l \int gdx\] |
