**Example 5**

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Example 5 Integrate the following functions 𝑤.𝑟.𝑡. 𝑥: (i) sin𝑚𝑥 sin 𝑚𝑥 𝑑𝑥 Let t = mx 𝑑𝑡𝑑𝑥=𝑚 𝑑𝑡𝑚=𝑑𝑥 Substituting values sin 𝑚𝑥 𝑑𝑥 = 𝑠𝑖𝑛𝑡× 𝑑𝑡𝑚 = 1𝑚 sin𝑡 𝑑𝑡 = 1𝑚 sin𝑡 𝑑𝑡 = 1𝑚 − cos𝑡+ C = − cos𝑡𝑚+ C Putting back value of t = m𝑥 = − 𝐜𝐨𝐬𝒎𝒙𝒎+ C Example 5 Integrate the following functions 𝑤.𝑟.𝑡. 𝑥: (ii) 2𝑥 sin 𝑥2+1 2𝑥 sin 𝑥2+1 dx Let t = 𝑥2+1 𝑑𝑡𝑑𝑥=2𝑥 dt = 2x dx Substituting 2𝑥 sin 𝑥2+1 𝑑𝑥 = sin𝑡 𝑑𝑡 = − cos𝑡+ C Putting value of t = 𝑥2+1 = − cos 𝒙𝟐+𝟏+ C Example 5 Integrate the following functions 𝑤.𝑟.𝑡. 𝑥: (iii) tan4 𝑥 sec2 𝑥 𝑥 tan4 𝑥 sec2 𝑥 𝑥 Let t = tan 𝑥 𝑑𝑡𝑑𝑥= 𝑠𝑒𝑐2 𝑥× 12 𝑥 2dt = dx 𝑠𝑒𝑐2 𝑥 Substituting value, 𝑡𝑎𝑛4 𝑥× 𝑠𝑒𝑐2 𝑥 𝑑𝑥 𝑥 = 𝑡4×2 𝑑𝑡 = 2 𝑡4 𝑑𝑡 = 2 𝑡55+ C Putting back value of t = 2 𝑡𝑎𝑛5 𝑥5+ C Example 5 Integrate the following functions 𝑤.𝑟.𝑡. 𝑥: (iv) sin tan−1𝑥1 + 𝑥2 sin tan−1𝑥1 + 𝑥2 Let t = 𝑡𝑎𝑛−1 𝑥 𝑑𝑡𝑑𝑥= 11+ 𝑥2 𝑑𝑡 = 𝑑𝑥1+ 𝑥2 Substituting sin tan−1𝑥1 + 𝑥2 = sin𝑡 𝑑𝑡 = − cos t + C Putting value of t = 𝑡𝑎𝑛−1 𝑥 = − cos (𝑡𝑎𝑛−1𝑥)+ C

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can contact him here.