Example 10 (ii) - Chapter 7 Class 12 Integrals

Transcript

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﷯