Check sibling questions

Ex 7.4, 18 - Integrate 5x - 2 / 1 + 2x + 3x2 - Ex 7.4

Ex 7.4, 18 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.4, 18 - Chapter 7 Class 12 Integrals - Part 3 Ex 7.4, 18 - Chapter 7 Class 12 Integrals - Part 4 Ex 7.4, 18 - Chapter 7 Class 12 Integrals - Part 5 Ex 7.4, 18 - Chapter 7 Class 12 Integrals - Part 6


Transcript

Ex 7.4, 18 5𝑥 − 2﷮1 + 2𝑥 + 3 𝑥﷮2﷯﷯ ﷮﷮ 5𝑥 − 2﷮1 + 2𝑥 + 3 𝑥﷮2﷯﷯﷯ 𝑑𝑥 =5 ﷮﷮ 𝑥 − 2﷮5﷯﷮1 + 2𝑥 + 3 𝑥﷮2﷯﷯﷯ 𝑑𝑥 = 5﷮6﷯ ﷮﷮ 6𝑥 − 12﷮5﷯﷮1 + 2𝑥 + 3 𝑥﷮2﷯﷯﷯ 𝑑𝑥 = 5﷮6﷯ ﷮﷮ 6𝑥 + 2 − 12﷮5﷯ − 2﷮1 + 2𝑥 + 3 𝑥﷮2﷯﷯﷯ 𝑑𝑥 = 5﷮6﷯ ﷮﷮ 6𝑥 + 2 − 22﷮5﷯﷮1 + 2𝑥 + 3 𝑥﷮2﷯﷯﷯ 𝑑𝑥 = 5﷮6﷯ ﷮﷮ 6𝑥 + 2﷮1 + 2𝑥 + 3 𝑥﷮2﷯﷯﷯ 𝑑𝑥− 5﷮6﷯× 22﷮5﷯ ﷮﷮ 𝑑𝑥﷮1 + 2𝑥 + 3 𝑥﷮2﷯﷯﷯ = 5﷮6﷯ ﷮﷮ 6𝑥 + 2﷮1 + 2𝑥 + 3 𝑥﷮2﷯﷯﷯ 𝑑𝑥− 11﷮3﷯ ﷮﷮ 𝑑𝑥﷮1 + 2𝑥 + 3 𝑥﷮2﷯﷯﷯ Solving 𝑰𝟏 I1 = 5﷮6﷯ ﷮﷮ 2 + 6𝑥﷮3 𝑥﷮2﷯+ 2𝑥 + 1﷯﷯ 𝑑𝑥 Let 3 𝑥﷮2﷯+2𝑥+1=𝑡 Diff both sides w.r.t.x 6𝑥+2+0= 𝑑𝑡﷮𝑑𝑥﷯ 𝑑𝑥= 𝑑𝑡﷮6𝑥 + 2﷯ Thus, our equation becomes I1= 5﷮6﷯ ﷮﷮ 2 + 6𝑥﷮3 𝑥﷮2﷯+ 2𝑥 + 1﷯﷯ 𝑑𝑥 Put the values of 3 𝑥﷮2﷯+2𝑥+1﷯ and 𝑑𝑥, we get I1= 5﷮6﷯ ﷮﷮ 6𝑥 + 2﷮𝑡﷯﷯ 𝑑𝑥 I1= 5﷮6﷯ ﷮﷮ 6𝑥 + 2﷮𝑡﷯﷯ × 𝑑𝑡﷮6𝑥 + 2﷯ I1= 5﷮6﷯ ﷮﷮ 1﷮𝑡﷯﷯ . 𝑑𝑡 I1= 5﷮6﷯ log﷮ 𝑡﷯﷯+𝐶1 I1= 5﷮6﷯ log﷮ 3 𝑥﷮2﷯+2𝑥+1﷯﷯+𝐶1 Solving 𝑰𝟐 I2= 11﷮3﷯ ﷮﷮ 1﷮3 𝑥﷮2﷯+ 2𝑥 + 1﷯﷯ . 𝑑𝑥 = 11﷮3﷯ ﷮﷮ 1﷮3 𝑥﷮2﷯+ 2𝑥﷮3﷯ + 1﷮3﷯﷯﷯﷯ . 𝑑𝑥 = 11﷮3 . 3﷯ ﷮﷮ 1﷮ 𝑥﷮2﷯+ 2𝑥﷮3﷯ + 1﷮3﷯﷯﷯ . 𝑑𝑥 = 11﷮9﷯ ﷮﷮ 1﷮ 𝑥﷮2﷯+ 2 𝑥﷯ 1﷮3﷯﷯ + 1﷮3﷯﷯﷯ . 𝑑𝑥 = 11﷮9﷯ ﷮﷮ 1﷮ 𝑥﷮2﷯+ 2 𝑥﷯ 1﷮3﷯﷯ + 1﷮3﷯﷯﷮2﷯− 1﷮3﷯﷯﷮2﷯+ 1﷮3﷯﷯﷯ . 𝑑𝑥 = 11﷮9﷯ ﷮﷮ 1﷮ 𝑥 + 1﷮3﷯﷯﷮2﷯− 1﷮3﷯﷯﷮2﷯+ 1﷮3﷯﷯﷯ . 𝑑𝑥 = 11﷮9﷯ ﷮﷮ 1﷮ 𝑥 + 1﷮3﷯﷯﷮2﷯− 1﷮9﷯ + 1﷮3﷯﷯﷯ . 𝑑𝑥 = 11﷮9﷯ ﷮﷮ 1﷮ 𝑥 + 1﷮3﷯﷯﷮2﷯ + 2﷮9﷯﷯﷯ . 𝑑𝑥 = 11﷮9﷯ ﷮﷮ 1﷮ 𝑥 + 1﷮3﷯﷯﷮2﷯ + ﷮2﷯﷮3﷯ ﷯﷮2﷯﷯﷯ . 𝑑𝑥 = 11﷮9﷯ 1﷮ ﷮2﷯﷮3﷯﷯ 𝑡𝑎𝑛﷮−1﷯ ﷮ 𝑥 + 1﷮3﷯﷮ ﷮2﷯﷮3﷯﷯﷯﷯+𝐶2 = 11﷮9﷯ 3﷮ ﷮2﷯﷯ 𝑡𝑎𝑛﷮−1﷯﷮ 3𝑥 +1﷮3﷯﷮ ﷮2﷯﷮3﷯﷯﷯﷯﷯+𝐶2 = 11﷮9﷯ 3﷮ ﷮2﷯﷯ 𝑡𝑎𝑛﷮−1﷯ ﷮ 3𝑥 + 1﷮ ﷮2﷯﷯﷯﷯﷯ = 11﷮9﷯ . 3﷮ ﷮2﷯﷯ 𝑡𝑎𝑛﷮−1﷯ ﷮ 3𝑥 + 1﷮ ﷮2﷯﷯﷯﷯ + 11﷮9﷯ 𝐶2 = 11﷮3 ﷮2﷯﷯ 𝑡𝑎𝑛﷮−1﷯ ﷮ 3𝑥 + 1﷮ ﷮2﷯﷯﷯﷯ + 𝐶3 Now, putting the values of I1 and I2 in (1) ﷮﷮ 5𝑥 − 2﷮1 + 2𝑥 + 3 𝑥﷮2﷯﷯﷯ . 𝑑𝑥 = 5﷮6﷯ ﷮﷮ 2 + 6𝑥﷮1 + 2𝑥 + 3 𝑥﷮2﷯﷯﷯ . 𝑑𝑥− 11﷮3﷯ ﷮﷮ 1﷮1 + 2𝑥 + 3 𝑥﷮2﷯﷯﷯ . 𝑑𝑥 = 𝟓﷮𝟔﷯ 𝒍𝒐𝒈﷮ 𝟑 𝒙﷮𝟐﷯+𝟐𝒙+𝟏﷯﷯− 𝟏𝟏﷮𝟑 ﷮𝟐﷯﷯ 𝒕𝒂𝒏﷮−𝟏﷯ ﷮ 𝟑𝒙 + 𝟏﷮ ﷮𝟐﷯﷯﷯﷯ + 𝑪

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.