Example 9 - Find integrals (i) dx / x2 - 6x + 13 - Class 12 - Examples

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Example 9 Find the following integrals: (i) ﷮﷮ 𝑑𝑥﷮ 𝑥﷮2﷯− 6𝑥 + 13﷯﷯ ﷮﷮ 𝑑𝑥﷮ 𝑥﷮2﷯− 6𝑥 + 13﷯﷯ = ﷮﷮ 𝑑𝑥﷮ 𝑥﷮2﷯− 2 × 3 × 𝑥 + 13﷯﷯ = ﷮﷮ 𝑑𝑥﷮ 𝑥﷮2﷯ − 2 . 3 𝑥 + 3﷮2﷯﷯ + 13 − 3﷮2﷯﷯﷯ = ﷮﷮ 𝑑𝑥﷮ 𝑥 − 3﷯﷮2﷯ + 13 − 9﷯﷯ = ﷮﷮ 𝑑𝑥﷮ 𝑥 − 3﷯﷮2﷯ + 4﷯﷯ = ﷮﷮ 𝑑𝑥﷮ 𝑥 − 3﷯﷮2﷯ + 2﷮2﷯﷯﷯ = 𝟏﷮𝟐﷯ 𝒕𝒂𝒏﷮−𝟏﷯﷮ 𝒙 − 𝟑﷮𝟐﷯﷯ +𝑪 Example 9 Find the following integrals: (ii) ﷮﷮ 𝑑𝑥﷮ 3𝑥﷮2﷯−13𝑥 + 10﷯﷯ ﷮﷮ 𝑑𝑥﷮ 3𝑥﷮2﷯ − 13𝑥 + 10﷯﷯ Solving denominator 3𝑥﷮2﷯+13𝑥−10 =3 𝑥﷮2﷯+ 13﷮3﷯𝑥 − 10﷮3﷯﷯ =3 𝑥﷮2﷯+2. 𝑥× 13﷮6﷯ − 10﷮3﷯﷯ Adding and subtracting 13﷮6﷯﷯﷮2﷯ =3 𝑥﷮2﷯+2. 𝑥× 13﷮6﷯+ 13﷮6﷯﷯﷮2﷯− 10﷮3﷯− 13﷮6﷯﷯﷮2﷯﷯ =3 𝑥+ 13﷮6﷯﷯﷮2﷯− 10﷮3﷯− 169﷮36﷯﷯﷯ =3 𝑥+ 13﷮6﷯﷯﷮2﷯− 10﷮3﷯ + 169﷮36﷯﷯﷯ =3 𝑥+ 13﷮6﷯﷯﷮2﷯− 120 + 69﷮36﷯ ﷯﷯ =3 𝑥+ 13﷮6﷯﷯﷮2﷯− 189﷮36﷯﷯ =3 𝑥+ 13﷮6﷯﷯﷮2﷯− 17﷮6﷯﷯﷮2﷯﷯ Hence, our equation becomes ﷮﷮ 𝑑𝑥﷮ 3𝑥﷮2﷯−13𝑥 + 10﷯﷯ = 1﷮3﷯ ﷮﷮ 𝑑𝑥﷮ 𝑥 + 13﷮6﷯﷯﷮2﷯− 17﷮6﷯﷯﷮2﷯﷯﷯ = 1﷮3﷯ × 1﷮2 17﷮6﷯﷯﷯ × log﷮ 𝑥 + 13﷮6﷯ − 17﷮6﷯﷮𝑥+ 13﷮6﷯ + 17﷮6﷯﷯﷯﷯ + C = 1﷮3﷯ × 6﷮2 17﷯﷯ × log﷮ 6𝑥 + 13 − 17﷮6﷯﷮ 6𝑥 +13 + 17﷮6﷯﷯﷯﷯ + C = 1﷮17﷯ log﷮ 6𝑥 − 4﷮6𝑥 + 30﷯﷯﷯ + C = 1﷮17﷯ log﷮ 2(3𝑥 − 2)﷮6(𝑥 + 5)﷯﷯﷯+ C = 1﷮17﷯ log﷮ 2(3𝑥 − 2)﷮6(𝑥 + 5)﷯﷯﷯+ C = 1﷮17﷯ log﷮ (3𝑥 − 2)﷮3(𝑥 + 5)﷯﷯﷯+ C = 1﷮17﷯ log﷮ (3𝑥 − 2)﷮(𝑥 + 5)﷯﷯﷯− 1﷮17﷯ log﷮3﷯ + C = 𝟏﷮𝟏𝟕﷯ 𝒍𝒐𝒈﷮ (𝟑𝒙 − 𝟐)﷮(𝒙 + 𝟓)﷯﷯﷯+ C1 Example 9 Find the following integrals: (iii) ﷮﷮ 𝑑𝑥﷮ ﷮5 𝑥﷮2﷯ − 2𝑥﷯ ﷯﷯ ﷮﷮ 𝑑𝑥﷮ ﷮5 𝑥﷮2﷯ − 2𝑥﷯ ﷯﷯ = ﷮﷮ 𝑑𝑥﷮ ﷮5 𝑥﷮2﷯ − 2﷮5﷯𝑥﷯﷯ ﷯﷯ = ﷮﷮ 𝑑𝑥﷮ ﷮5 𝑥﷮2﷯ − 2 𝑥﷯ 1﷮5﷯﷯﷯﷯ ﷯﷯ = ﷮﷮ 𝑑𝑥﷮ ﷮5 𝑥﷮2﷯ − 2 𝑥﷯ 1﷮5﷯﷯ + 1﷮5﷯﷯﷮2﷯− 1﷮5﷯﷯﷮2﷯﷯﷯ ﷯﷯ = ﷮﷮ 𝑑𝑥﷮ ﷮5 𝑥 − 1﷮5﷯﷯﷮2﷯− 1﷮5﷯﷯﷮2﷯﷯﷯ ﷯﷯ = ﷮﷮ 𝑑𝑥﷮ ﷮5﷯ ﷮ 𝑥 − 1﷮5﷯﷯﷮2﷯− 1﷮5﷯﷯﷮2﷯﷯﷯﷯ = ﷮﷮ 𝑑𝑥﷮ ﷮5﷯ ﷮ 𝑥 − 1﷮5﷯﷯﷮2﷯− 1﷮5﷯﷯﷮2﷯﷯﷯﷯ = 1﷮ ﷮5﷯﷯ 𝑙𝑜𝑔 𝑥− 1﷮5﷯+ ﷮ 𝑥− 1﷮5﷯﷯﷮2﷯− 1﷮5﷯﷯﷮2﷯﷯﷯+𝐶1﷯ = 1﷮ ﷮5﷯﷯𝑙𝑜𝑔 𝑥− 1﷮5﷯+ ﷮ 𝑥− 1﷮5﷯﷯﷮2﷯− 1﷮5﷯﷯﷮2﷯﷯﷯+ 𝐶1﷮ ﷮5﷯﷯ = 1﷮ ﷮5﷯﷯𝑙𝑜𝑔 𝑥− 1﷮5﷯+ ﷮ 𝑥﷮2﷯+ 1﷮5﷯﷯﷮2﷯−2 𝑥﷯ 1﷮5﷯﷯− 1﷮5﷯﷯﷮2﷯﷯﷯+𝐶 = 1﷮ ﷮5﷯﷯𝑙𝑜𝑔 𝑥− 1﷮5﷯+ ﷮ 𝑥﷮2﷯− 2𝑥﷮5﷯﷯﷯+𝐶

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