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Chapter 7 Class 12 Integrals
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Example 10 (ii) - Example 10Find the following integrals x + 3 /root 5 - 4x - x2 dx - Chapter 7 Class 12 CBSE NCERT Math - Integration by specific formulaes - Method 10

Example 10 (ii) - Chapter 7 Class 12 Integrals - Part 2
Example 10 (ii) - Chapter 7 Class 12 Integrals - Part 3 Example 10 (ii) - Chapter 7 Class 12 Integrals - Part 4 Example 10 (ii) - Chapter 7 Class 12 Integrals - Part 5 Example 10 (ii) - Chapter 7 Class 12 Integrals - Part 6

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Example 10 Find the following integrals: (ii) ﷮﷮ 𝑥 + 3﷮ ﷮5 − 4𝑥 − 𝑥﷮2﷯﷯﷯﷯ 𝑑𝑥 It can be written in the form 𝑥+3= A 𝑑﷮𝑑𝑥﷯ − 𝑥﷮2﷯−4𝑥+5﷯﷯+ B 𝑥+3= A −2𝑥−4﷯+ B 𝑥+3= 2A﷯ 𝑥﷯−4A+B Finding A & B Now, we know that 𝑥+3=A −2𝑥−4﷯+B 𝑥+3= −1﷮ 2﷯ −2𝑥−4﷯+1 Step 2 : Integrating ﷮﷮ 𝑥+3﷮ ﷮5 − 4𝑥 + 𝑥﷮2﷯﷯﷯﷯.𝑑𝑥= ﷮﷮ 1﷮2﷯ −2𝑥 − 4﷯+1﷮ ﷮5 − 4𝑥 − 𝑥﷮2﷯﷯﷯﷯.𝑑𝑥 = ﷮﷮ −1﷮ 2﷯ −2𝑥 − 4﷯+1﷮ ﷮5 − 4𝑥 + 𝑥﷮2﷯﷯﷯﷯𝑑𝑥+ ﷮﷮ 1﷮ ﷮5 − 4𝑥 − 𝑥﷮2﷯﷯﷯﷯𝑑𝑥 = −1﷮ 2﷯ ﷮﷮ −2𝑥 − 4﷮ ﷮5 − 4𝑥 + 𝑥﷮2﷯﷯﷯﷯.𝑑𝑥+ ﷮﷮ 1﷮ ﷮5 − 4𝑥 − 𝑥﷮2﷯﷯﷯﷯.𝑑𝑥 Taking I1 I1 = −1﷮ 2﷯ ﷮﷮ −2𝑥 − 4﷮ ﷮− 𝑥﷮2﷯ − 4𝑥 + 5﷯﷯.𝑑𝑥﷯ Let t = − 𝑥﷮2﷯ − 4𝑥 + 5 Differentiating both sides w.r.t. 𝑥 −2𝑥 − 4 = 𝑑𝑡﷮𝑑𝑥﷯ 𝑑𝑥= 𝑑𝑡﷮− 2𝑥 − 4﷯ Now, I1 = −1﷮ 2﷯ ﷮﷮ −2𝑥 − 4﷮ ﷮− 𝑥﷮2﷯ − 4𝑥 + 5﷯﷯.𝑑𝑥﷯ Putting the values of 𝑥﷮2﷯−4𝑥+5﷯ and 𝑑𝑥 I1 = −1﷮ 2﷯ ﷮﷮ −2𝑥 − 4﷮ ﷮𝑡﷯﷯. 𝑑𝑡﷮ −2𝑥 −4﷯﷯﷯ I1 = −1﷮ 2﷯ ﷮﷮ 1﷮ ﷮𝑡﷯﷯.𝑑𝑡﷯ I1 = −1﷮ 2﷯ ﷮﷮ 1﷮ 𝑡﷯﷮ 1﷮2﷯﷯﷯.𝑑𝑡﷯ I1 = −1﷮ 2﷯ ﷮﷮ 𝑡﷯﷮ −1﷮2﷯﷯ 𝑑𝑡﷯ I1 = −1﷮ 2﷯ 𝑡﷮ −1﷮ 2﷯ + 1﷯﷮ −1﷮ 2﷯ + 1﷯﷯+𝐶1 I1 = −1﷮ 2﷯ 𝑡﷮ 1﷮ 2﷯ ﷯﷮ 1﷮2﷯﷯﷯+𝐶1 I1 = − 𝑡﷮ 1﷮ 2﷯ ﷯+𝐶1 I1 = − ﷮𝑡﷯+𝐶1 I1 = − ﷮− 𝑥﷮2﷯−4𝑥+5﷯+𝐶1 I1 = − ﷮5−4𝑥− 𝑥﷮2﷯﷯+𝐶1 Taking 𝐈𝟐 I2 = ﷮﷮ 1﷮ ﷮5 − 4𝑥 − 𝑥﷮2﷯﷯﷯.𝑑𝑥﷯ I2 = ﷮﷮ 1﷮ ﷮− 𝑥﷮2﷯ + 4𝑥 − 5﷯﷯﷯𝑑𝑥﷯ I2 = ﷮﷮ 1﷮ ﷮− 𝑥﷮2﷯ + 2 2﷯ 𝑥﷯ − 5﷯﷯﷯𝑑𝑥﷯ I2 = ﷮﷮ 1﷮ ﷮− 𝑥﷮2﷯ + 2 2﷯ 𝑥﷯ + 2﷯﷮2﷯ − 2﷯﷮2﷯ − 5﷯﷯﷯𝑑𝑥﷯ I2 = ﷮﷮ 1﷮ ﷮− 𝑥+2﷯﷮2﷯ − 4 − 5﷯﷯﷯𝑑𝑥﷯ I2 = ﷮﷮ 1﷮ ﷮− 𝑥+2﷯﷮2﷯ − 9﷯﷯﷯𝑑𝑥﷯ I2 = ﷮﷮ 1﷮ ﷮9 − 𝑥 + 2﷯﷮2﷯﷯﷯𝑑𝑥﷯ I2 = ﷮﷮ 1﷮ ﷮ 3﷯﷮2﷯− 𝑥+2﷯﷮2﷯﷯﷯𝑑𝑥﷯ I2 = sin﷮−1﷯﷮ 𝑥 + 2﷮3﷯﷯+𝐶2﷯ Now, putting the values of I1 and I2 in eq. (1) we get ﷮﷮ 𝑥 + 3﷮ ﷮5 − 4𝑥 − 𝑥﷮2﷯﷯﷯.𝑑𝑥= −1﷮ 2﷯ ﷮﷮ − 2𝑥 − 4﷮ ﷮5 − 4𝑥 − 𝑥﷮2﷯﷯﷯﷯﷯+ ﷮﷮ 1﷮ ﷮5 − 4𝑥 − 𝑥﷮2﷯﷯﷯﷯𝑑𝑥 =− ﷮5 − 4𝑥 − 𝑥﷮2﷯﷯+𝐶1+ sin﷮−1﷯﷮ 𝑥 + 2﷮3﷯﷯+𝐶2﷯ =− ﷮𝟓 − 𝟒𝒙 − 𝒙﷮𝟐﷯﷯+ 𝒔𝒊𝒏﷮−𝟏﷯﷮ 𝒙 + 𝟐﷮𝟑﷯﷯+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.