# Ex 7.10, 6

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Ex7.10, 6 Evaluate the integrals using substitution 02 𝑑𝑥𝑥 + 4 − 𝑥2 We can write 02 𝑑𝑥𝑥 + 4 − 𝑥2= 02 𝑑𝑥− 𝑥2 − 𝑥 − 4 =− 02 𝑑𝑥 𝑥2 − 𝑥 − 4 =− 02 𝑑𝑥 𝑥2 −2 × 12 × 𝑥 − 4 =− 02 𝑑𝑥 𝑥2 −2 × 12 × 𝑥 + 1 22 − 1 22 − 4 =− 02 𝑑𝑥 𝑥 − 122− 14 − 4 =− 02 𝑑𝑥 𝑥 − 122− 174 =− 02 𝑑𝑥 𝑥 − 122− 1742 Let 𝑡=𝑥− 12 Differentiating w.r.t.𝑥 𝑑𝑡𝑑𝑥=1 𝑑𝑡=𝑑𝑥 When x varies from 0 to 2, then t varies from −12 to 32. Therefore, − 02 𝑑𝑥 𝑥 − 122− 1722=− −12 32 𝑑𝑡𝑡 − 1722 =− 12 172𝑙𝑜𝑔 𝑡 − 172𝑡 + 172 −1 2 32 =− 1 17 𝑙𝑜𝑔 32 − 172 32 + 172+𝑙𝑜𝑔 −1 2 − 172 −1 2 + 172 =− 1 17 𝑙𝑜𝑔 3 − 173 + 17+𝑙𝑜𝑔 − 1 + 17− 1 − 17 =− 1 17𝑙𝑜𝑔 3 − 173 + 17 1 + 171 − 17 =− 1 17𝑙𝑜𝑔 3 − 173 + 17 × 1 − 171 + 17 =− 1 17𝑙𝑜𝑔 3+17 − 3 17 − 173 +17 + 3 17 + 17 =− 1 17𝑙𝑜𝑔 20 − 4 1720 + 4 17 =− 1 17𝑙𝑜𝑔 4 5 − 174 5 + 17 =− 1 17𝑙𝑜𝑔 5 − 175 + 17 = 1 17𝑙𝑜𝑔 5 − 175 + 17 −1 = 1 17𝑙𝑜𝑔 5 + 175 − 17 = 1 17𝑙𝑜𝑔 5 + 175 − 17 × 5 + 175 + 17 = 1 17𝑙𝑜𝑔 5 − 172 52 − 172 = 1 17𝑙𝑜𝑔 25 + 17 + 10 1725 − 17 = 1 17𝑙𝑜𝑔 42 + 10 178 = 𝟏 𝟏𝟕𝒍𝒐𝒈 𝟐𝟏 + 𝟓 𝟏𝟕𝟒

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Davneet Singh

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