Example 27 (ii) - Evaluate the following integrals root x / 30 -x 3/2 dx - Definate Integration - By Substitution

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  1. Chapter 7 Class 12 Integrals
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Example 27 Evaluate the following integrals: (ii) ﷐4﷮9﷮﷐﷐﷮𝑥﷯﷮﷐﷐30 − ﷐𝑥﷮﷐3﷮2﷯﷯﷯﷮2﷯ ﷯﷯ 𝑑𝑥 Step 1 :- ﷐﷮﷮﷐﷐﷮𝑥﷯﷮﷐﷐30 − ﷐𝑥﷮﷐3﷮2﷯﷯﷯﷮2﷯﷯﷯ 𝑑𝑥 Let 30−﷐𝑥﷮﷐3﷮2﷯﷯=𝑡 Differentiating w.r.t. 𝑥 both sides ﷐𝑑﷐30 − ﷐𝑥﷮﷐3﷮2﷯﷯﷯﷮𝑑𝑥﷯=﷐𝑑𝑡﷮𝑑𝑥﷯ −﷐3﷮2﷯﷐𝑥﷮﷐3﷮2﷯ −1﷯=﷐𝑑𝑡﷮𝑑𝑥﷯ −﷐3﷮2﷯﷐𝑥﷮﷐1﷮2﷯ ﷯=﷐𝑑𝑡﷮𝑑𝑥﷯ 𝑑𝑥=﷐𝑑𝑡﷮− ﷐3﷮2﷯﷐ 𝑥﷮ ﷐1﷮2﷯ ﷯﷯ 𝑑𝑥=﷐−2𝑑𝑡﷮3﷐﷮𝑥﷯﷯ Therefore, our equation becomes ﷐﷮﷮﷐﷐﷮𝑥﷯ 𝑑𝑥 ﷮﷐﷐30−﷐𝑥﷮ ﷐3﷮2﷯﷯﷯﷮2﷯﷯=﷐﷮﷮﷐﷐﷮𝑥﷯﷮﷐𝑡﷮2﷯﷯ ﷐−2 𝑑𝑡﷮3 ﷐﷮𝑥﷯﷯﷯﷯ =﷐−2﷮ 3﷯﷐﷮﷮﷐ 𝑑𝑡﷮﷐𝑡﷮2﷯﷯﷯ =﷐−2﷮ 3﷯﷐﷮﷮﷐𝑡﷮2﷯ 𝑑𝑡﷯ =﷐−2﷮ 3﷯ ﷐﷐𝑡﷮− 2 + 1﷯﷮− 2 + 1﷯ =﷐−2﷮ 3﷯ ﷐﷐𝑡﷮− 1﷯﷮−1﷯ =﷐2﷮3﷯ ﷐𝑡﷮−1﷯ =﷐2﷮3𝑡﷯ Putting 𝑡=﷐30−﷐𝑥﷮﷐3﷮2﷯﷯﷯ =﷐2﷮3﷐30 − ﷐𝑥﷮﷐3﷮2﷯﷯﷯﷯ Hence F﷐𝑥﷯=﷐2﷮3﷐30 − ﷐𝑥﷮﷐3﷮2﷯﷯﷯﷯ Step 2 :- ﷐4﷮9﷮﷐﷐﷮𝑥﷯﷮﷐30 − ﷐𝑥﷮ ﷐3﷮2﷯﷯﷯﷯﷯ 𝑑𝑥=𝐹﷐9﷯−𝐹﷐4﷯ =﷐2﷮3﷐30 − ﷐﷐9﷯﷮﷐3﷮2﷯﷯﷯﷯−﷐2﷮3﷐30 − ﷐﷐4﷯﷮﷐3﷮2﷯﷯﷯﷯ = ﷐2﷮3 ﷐30 − ﷐﷐﷐3﷮2﷯﷯﷮﷐2﷮3﷯﷯﷯ ﷯−﷐2﷮3 ﷐30 − ﷐﷐﷐2﷮2﷯﷯﷮﷐2﷮3﷯﷯﷯﷯ =﷐2﷮3﷯﷐﷐1﷮30 − ﷐3﷮3﷯﷯−﷐1﷮30 − ﷐2﷮3﷯﷯﷯ =﷐2﷮3﷯﷐﷐1﷮30 − 27﷯−﷐1﷮30 − 8﷯﷯ =﷐2﷮3﷯﷐﷐1﷮3﷯−﷐1﷮22﷯﷯ =﷐2﷮3﷯﷐﷐22 − 3﷮3 × 22﷯﷯ =﷐2﷮3﷯ ﷐﷐19﷮66﷯﷯ =﷐19﷮3 (33)﷯ =﷐𝟏𝟗﷮𝟗𝟗﷯

Chapter 7 Class 12 Integrals
Concept wise

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