Ex 7.2, 14 - Chapter 7 Class 12 Integrals

Transcript

Ex 7.2, 14 Integrate the function: 1/(𝑥(log⁡𝑥 )^𝑚 ) , 𝑥 > 0 Step 1: Let log⁡𝑥=𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 1/𝑥= 𝑑𝑡/𝑑𝑥 𝑑𝑥=𝑥 . 𝑑𝑡 Step 2: Integrating the function ∫1▒〖" " 1/(𝑥(log⁡𝑥 )^𝑚 )〗 . 𝑑𝑥 Putting 𝑙𝑜𝑔⁡𝑥=𝑡 & 𝑑𝑥=𝑥 . 𝑑𝑡 = ∫1▒〖" " 1/(𝑥 . 𝑡^𝑚 )〗 . 𝑥 𝑑𝑡 = ∫1▒〖" " 1/𝑡^𝑚 〗 . 𝑑𝑡 = ∫1▒〖" " 𝑡^(−𝑚) 〗 . 𝑑𝑡 = 𝑡^(−𝑚 + 1)/(−𝑚 +1) +𝐶 = 𝑡^(1 − 𝑚)/(1 − 𝑚) +𝐶 Putting back t = log x = (𝒍𝒐𝒈⁡𝒙 )^(𝟏 − 𝒎)/(𝟏 − 𝒎) +𝑪 (Using ∫1▒𝑥^𝑛 . 𝑑𝑥=𝑥^(𝑛+1)/(𝑛 +1))