Examples

Chapter 7 Class 12 Integrals
Serial order wise

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

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Example 6 Find the following integrals (iii) ∫1▒1/(1 + tan⁡𝑥 ) 𝑑𝑥 The given function cannot be integrated by direct substitution, so we have to simplify the given function . Simplifying the given function and integrating. ∫1▒1/(1 + tan⁡𝑥 ) .𝑑𝑥 = ∫1▒1/(1 + sin⁡𝑥/cos⁡𝑥 ) .𝑑𝑥 = ∫1▒1/((cos⁡𝑥 + sin⁡𝑥)/cos⁡𝑥 ) .𝑑𝑥 (𝑈𝑠𝑖𝑛𝑔 tan⁡𝑥=sin⁡𝑥/cos⁡𝑥 ) = ∫1▒cos⁡𝑥/(sin⁡𝑥 + cos⁡𝑥 ) .𝑑𝑥 Multiplying and dividing by 2 = ∫1▒(2 cos⁡𝑥)/(2 (sin⁡𝑥 + cos⁡𝑥 ) ) .𝑑𝑥 = ∫1▒(cos⁡𝑥 + cos⁡𝑥)/(2 (sin⁡𝑥 + cos⁡𝑥 ) ) .𝑑𝑥 Adding and subtracting sin⁡𝑥 in the numerator = ∫1▒(cos⁡𝑥 + cos⁡𝑥 + sin⁡𝑥 − sin⁡𝑥)/(2 (sin⁡𝑥 + cos⁡𝑥 ) ) .𝑑𝑥 = 1/2 ∫1▒(sin⁡𝑥 + cos⁡𝑥 + cos⁡𝑥 − sin⁡𝑥)/(sin⁡𝑥 + cos⁡𝑥 ) .𝑑𝑥 = 1/2 ∫1▒[(sin⁡𝑥 + cos⁡𝑥)/(sin⁡𝑥 + cos⁡𝑥 )+〖cos⁡𝑥 − sin〗⁡𝑥/(sin⁡𝑥 + cos⁡𝑥 )] 𝑑𝑥 = 1/2 ∫1▒[1+〖cos⁡𝑥 − sin〗⁡𝑥/(sin⁡𝑥 + cos⁡𝑥 )] 𝑑𝑥 = 1/2 [∫1▒〖1.𝑑𝑥+∫1▒〖cos⁡𝑥 − sin〗⁡𝑥/(sin⁡𝑥 + cos⁡𝑥 )〗 𝑑𝑥] = 1/2 [𝑥+𝐶1+∫1▒〖cos⁡𝑥 − sin〗⁡𝑥/(sin⁡𝑥 + cos⁡𝑥 ) 𝑑𝑥] Take, I1 =∫1▒〖cos⁡𝑥 − sin〗⁡𝑥/(sin⁡𝑥 + cos⁡𝑥 ) .𝑑𝑥 Let sin⁡𝑥 + cos⁡𝑥=𝑡 Differentiate both sides 𝑤.𝑟.𝑡.𝑥. cos⁡𝑥−sin⁡𝑥=𝑑𝑡/𝑑𝑥 𝑑𝑥=𝑑𝑡/(cos⁡𝑥 − sin⁡𝑥 ) Putting value of (𝑠𝑖𝑛⁡𝑥+𝑐𝑜𝑠⁡𝑥 ) and 𝑑𝑥 in I1 . I1=∫1▒〖cos⁡𝑥 − sin〗⁡𝑥/(sin⁡𝑥 + cos⁡𝑥 ) .𝑑𝑥 I1 = ∫1▒〖cos⁡𝑥 − sin〗⁡𝑥/𝑡 .𝑑𝑡/(cos⁡𝑥 − sin⁡𝑥 ) I1 =∫1▒1/𝑡 .𝑑𝑡 Let sin⁡𝑥 + cos⁡𝑥=𝑡 Differentiate both sides 𝑤.𝑟.𝑡.𝑥. cos⁡𝑥−sin⁡𝑥=𝑑𝑡/𝑑𝑥 𝑑𝑥=𝑑𝑡/(cos⁡𝑥 − sin⁡𝑥 ) Putting value of (𝑠𝑖𝑛⁡𝑥+𝑐𝑜𝑠⁡𝑥 ) and 𝑑𝑥 in I1 . I1=∫1▒〖cos⁡𝑥 − sin〗⁡𝑥/(sin⁡𝑥 + cos⁡𝑥 ) .𝑑𝑥 I1 = ∫1▒〖cos⁡𝑥 − sin〗⁡𝑥/𝑡 .𝑑𝑡/(cos⁡𝑥 − sin⁡𝑥 ) I1 =∫1▒1/𝑡 .𝑑𝑡 (𝑈𝑠𝑖𝑛𝑔 ∫1▒〖1/𝑥 .〗 𝑑𝑥=log⁡〖 |𝑥|〗+𝐶) (𝑈𝑠𝑖𝑛𝑔 𝑡=sin⁡𝑥+cos⁡𝑥 ) (𝑊ℎ𝑒𝑟𝑒 𝐶=𝐶1/2+𝐶2/2)