Example 27 (iii) - Example 27
Evaluate the following integrals:
(iii) ﷐1﷮2﷮﷐𝑥 𝑑𝑥﷮﷐𝑥 + 1﷯ ﷐𝑥 + 2﷯﷯﷯ 𝑑𝑥  
Step  - Definate Integration - By Partial Fraction

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  1. Chapter 7 Class 12 Integrals
  2. Concept wise
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Example 27 Evaluate the following integrals: (iii) ﷐1﷮2﷮﷐𝑥 𝑑𝑥﷮﷐𝑥 + 1﷯ ﷐𝑥 + 2﷯﷯﷯ 𝑑𝑥 Step 1 :- 𝐹﷐𝑥﷯=﷐﷮﷮﷐𝑥 𝑑𝑥﷮﷐𝑥 + 1﷯﷐𝑥 + 2﷯﷯﷯ We can write the integrate as : ﷐𝑥﷮﷐𝑥 + 1﷯﷐𝑥 + 2﷯﷯=﷐A﷮𝑥 + 1﷯+﷐B﷮𝑥 + 2﷯ ﷐𝑥﷮﷐𝑥 + 1﷯﷐𝑥 + 2﷯﷯=﷐A﷐𝑥 + 2﷯ + B﷐𝑥 + 1﷯﷮﷐𝑥 + 1﷯﷐𝑥 + 2﷯﷯ By canceling denominators 𝑥=A﷐𝑥+2﷯+B﷐𝑥+1﷯ Therefore ﷐𝑥﷮﷐𝑥 + 1﷯﷐𝑥 + 2﷯﷯=﷐−1﷮𝑥 + 1﷯+﷐2﷮𝑥 + 2﷯ Integrating w.r.t.𝑥 ﷐﷮﷮﷐𝑥﷮﷐𝑥+1﷯﷐𝑥+2﷯﷯=﷐﷮﷮﷐−1﷮﷐𝑥+1﷯﷯𝑑𝑥﷯+﷐﷮﷮﷐2﷮𝑥+2﷯𝑑𝑥﷯﷯ =−𝑙𝑜𝑔﷐𝑥+1﷯+2𝑙𝑜𝑔﷐𝑥+2﷯ =−𝑙𝑜𝑔﷐𝑥+1﷯+𝑙𝑜𝑔﷐﷐𝑥+2﷯﷮2﷯ =𝑙𝑜𝑔﷐﷐𝑥+2﷯﷮2﷯−𝑙𝑜𝑔﷐𝑥+1﷯ =𝑙𝑜𝑔﷐﷐﷐﷐𝑥 + 2﷯﷮2﷯﷮𝑥 + 1﷯﷯ Hence 𝐹﷐𝑥﷯=𝑙𝑜𝑔﷐﷐﷐﷐𝑥 + 2﷯﷮2﷯﷮𝑥 + 1﷯﷯ Step 2 :- ﷐1﷮2﷮﷐𝑥﷮﷐𝑥 + 1﷯﷐𝑥 + 2﷯﷯𝑑𝑥﷯=𝐹﷐2﷯−𝐹﷐1﷯ ﷐1﷮2﷮﷐𝑥﷮﷐𝑥 + 1﷯﷐𝑥 + 2﷯﷯𝑑𝑥﷯=𝑙𝑜𝑔﷐﷐﷐﷐2 + 2﷯﷮2﷯﷮2 + 1﷯﷯−𝑙𝑜𝑔﷐﷐﷐﷐1 + 2﷯﷮2﷯﷮1 + 1﷯﷯ =𝑙𝑜𝑔﷐﷐﷐﷐4﷯﷮2﷯﷮3﷯﷯−𝑙𝑜𝑔﷐﷐﷐﷐3﷯﷮2﷯﷮2﷯﷯ =𝑙𝑜𝑔﷐﷐﷐﷐4﷮2﷯﷮3﷯﷮﷐﷐3﷮2﷯﷮2﷯﷯﷯ =𝑙𝑜𝑔﷐﷐﷐4﷮2﷯﷮3﷯ × ﷐2﷮﷐3﷮2﷯﷯﷯ =𝑙𝑜𝑔﷐﷐16﷮3﷯ × ﷐2﷮9﷯﷯ =𝑙𝑜𝑔﷐﷐32﷮27﷯ ﷯ =﷐𝐥𝐨𝐠﷮﷐﷐𝟑𝟐﷮𝟐𝟕﷯﷯﷯

Chapter 7 Class 12 Integrals
Concept wise

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