Examples

Chapter 7 Class 12 Integrals
Serial order wise

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class

### Transcript

Example 27 (Method 1) Evaluate ﷐0﷮1﷮﷐﷐﷐tan﷮−1﷯﷮𝑥﷯﷮1 + ﷐𝑥﷮2﷯﷯﷯ 𝑑𝑥 Step 1 : Let F﷐𝑥﷯=﷐﷮﷮﷐﷐﷐tan﷮−1﷯﷮𝑥﷯﷮1+﷐ 𝑥﷮2﷯﷯﷯𝑑𝑥 Put ﷐﷐tan﷮−1﷯﷮𝑥﷯=𝑡 Differentiating w.r.t.𝑥 ﷐𝑑﷮𝑑𝑥﷯﷐﷐﷐tan﷮−1﷯﷮𝑥﷯﷯=﷐𝑑𝑡﷮𝑑𝑥﷯ ﷐1﷮1 + ﷐𝑥﷮2﷯﷯=﷐𝑑𝑡﷮𝑑𝑥﷯ Therefore, ﷐﷮﷮﷐﷐﷐tan﷮−1﷯﷮𝑥﷯﷮1+﷐ 𝑥﷮2﷯﷯﷯𝑑𝑥=﷐﷮﷮﷐𝑡﷮1+﷐𝑥﷮2﷯﷯ × ﷐1+﷐𝑥﷮2﷯﷯𝑑𝑡﷯ =﷐﷮﷮𝑡 𝑑𝑡﷯ =﷐﷐𝑡﷮2﷯﷮2﷯ Putting 𝑡=﷐﷐𝑡𝑎𝑛﷮−1﷯﷮𝑥﷯ =﷐﷐﷐﷐﷐tan﷮−1﷯﷮𝑥﷯﷯﷮2﷯﷮2﷯ Hence 𝐹﷐𝑥﷯=﷐﷐﷐﷐﷐tan﷮−1﷯﷮𝑥﷯﷯﷮2﷯﷮2﷯ Step 2 : ﷐﷮﷮﷐﷐﷐𝑡𝑎𝑛﷮−1﷯﷮𝑥﷯﷮1 + ﷐𝑥﷮2﷯﷯﷯=𝐹﷐1﷯−F﷐0﷯ =﷐1﷮2﷯﷐﷐﷐﷐tan﷮−1﷯﷮1﷯﷯﷮2﷯ −﷐1﷮2﷯﷐﷐﷐﷐tan﷮−1﷯﷮0﷯﷯﷮2﷯ =﷐1﷮2﷯﷐﷐﷐𝜋﷮4﷯﷯﷮2﷯−﷐1﷮2﷯﷐﷐0﷯﷮2﷯ =﷐1﷮2﷯ ﷐﷐𝜋﷮2﷯﷮16﷯ = ﷐﷐𝝅﷮𝟐﷯﷮𝟑𝟐﷯ Example 27 (Method 2) Evaluate ﷐0﷮1﷮﷐﷐﷐tan﷮−1﷯﷮𝑥﷯﷮1 + ﷐𝑥﷮2﷯﷯﷯ 𝑑𝑥 Put 𝑡=﷐﷐tan﷮−1﷯﷮𝑥﷯ Differentiating w.r.t.𝑥 ﷐𝑑𝑡﷮𝑑𝑥﷯=﷐𝑑﷮𝑑𝑥﷯﷐﷐﷐tan﷮−1﷯﷮𝑥﷯﷯ ﷐𝑑𝑡﷮𝑑𝑥﷯=﷐1﷮1 + ﷐𝑥﷮2﷯﷯ ﷐1+﷐𝑥﷮2﷯﷯𝑑𝑡=𝑑𝑥 Hence when value of x varies from 0 to 1, value of t varies from 0 to ﷐𝜋﷮4﷯ Therefore, ﷐0﷮1﷮﷐﷐﷐tan﷮−1﷯﷮𝑥﷯﷮1 + ﷐𝑥﷮2﷯﷯﷯=﷐0﷮﷐𝜋﷮4﷯﷮﷐𝑡﷮1 + ﷐𝑥﷮2﷯﷯﷯𝑑𝑥 ﷐1+﷐𝑥﷮2﷯﷯𝑑𝑡 =﷐0﷮﷐𝜋﷮4﷯﷮ 𝑡 𝑑𝑡﷯ =﷐﷐﷐﷐𝑡﷮2﷯﷮2﷯﷯﷮0﷮﷐𝜋﷮4﷯﷯ =﷐1﷮2﷯﷐﷐﷐﷐𝜋﷮4﷯﷯﷮2﷯−﷐﷐0﷯﷮2﷯﷯ =﷐1﷮2﷯ × ﷐﷐𝜋﷮2﷯﷮16﷯ = ﷐﷐𝝅﷮𝟐﷯﷮𝟑𝟐﷯

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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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.