Data Integration

Data integration involves combining data residing in different sources and providing users with a unified view of these data. This process becomes significant in a variety of situations both commercial  and scientific. Data integration appears with increasing frequency as the volume and the need to share existing data explodes.We will see some example problems for data integration below.

(Source: Wiki).

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General Formula for Data Integration

∫ k dx = kx + c

∫ xn dx = x(n + 1) / (n + 1) + c

∫ udv = uv - v ∫ du

Data Integration - Examples

solved example problems
Example 1: Integrate the given data value ∫ (16x + 9x6 - 10x2) dx

Solution:

Integrate the given data with respect to x, we get

∫ (16x + 9x6 - 10x2) dx =  ∫ 16x dx +  ∫ 9x6 dx -  ∫ 10x2 dx

= 16 (x2 / 2) + 8 (x7 / 7) - 9 (x3 / 3)

= 8x2 + (8/7)x7 - 3x3 + C

Answer: 8x2 + (8/7)x7 - 3x3 + C

Example 2: Integrate the given data ∫ sin 9x sin 5x dx

Solution:

Rearrange the given data,

sin A sin B = (cos (A - B) - cos (A + B)) / 2

From given data, A = 9x, B = 5x

sin 9x sin 5x =  1/2 * cos 4x - cos 14x

Substitute the above values in the given data, we get

Therefore,

∫ sin 9x sin 5x dx = ∫ 1/2 * cos 4x - cos 14x dx

= 1/2 [ ∫ cos 4x dx -  ∫ cos 14x dx]

= 1/2 * [ sin 4x/4 - sin 14x/14)] + C

= sin 4x/8 - sin 12x/28 + C

Answer: sin 4x/8 - sin 12x/28 + C

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Data Integration - Practice Problem

Problem: Integrate the given data ∫ (20x2 - 6x6 + 63x) dx

Answer: (20/3)x3 - (6/7)x7 + 63x2 + C