Misc 22 - Chapter 7 Class 12 Integrals

Last updated at August 8, 2026 by

Β  Misc 22 - Integrate tan-1 root (1 - x) / (1 + x) - Miscellaneous - Miscellaneous

part 2 - Misc 22 - Miscellaneous - Serial order wise - Chapter 7 Class 12 Integrals
part 3 - Misc 22 - Miscellaneous - Serial order wise - Chapter 7 Class 12 Integrals

Take a fresh quiz. Then take another.
Every attempt is a new AI-adaptive Teachoo quiz with 3 questions, selected from your answers, mistakes, and progress.
Remove Ads

Transcript

Misc 22 Integrate the function tan^(βˆ’1)⁑√((1 βˆ’ π‘₯)/(1 + π‘₯)) Let x = cos 2πœƒ 𝑑π‘₯/π‘‘πœƒ=βˆ’2 sin⁑〖2πœƒ γ€— dx = βˆ’2 sin 2πœƒ dπœƒ Substituting, ∫1β–’γ€–tan^(βˆ’1)⁑√((1 βˆ’ π‘₯)/(1 + π‘₯)) 𝑑π‘₯γ€— = ∫1β–’γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) √((1 βˆ’ cos⁑2πœƒ)/(1 + cos⁑2πœƒ ))Γ—(βˆ’2 sin⁑〖2 πœƒ)γ€— 𝑑 πœƒ = βˆ’2∫1β–’γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) √((1 βˆ’ (1 βˆ’ 2〖𝑠𝑖𝑛〗^2 πœƒ))/(1 + (2γ€–π‘π‘œπ‘ γ€—^2 πœƒ βˆ’ 1) ))Γ—("sin 2πœƒ dπœƒ " ) = βˆ’2∫1β–’γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) √((sin^2β‘πœƒ/cos^2β‘πœƒ ) )Γ—sin⁑〖2πœƒ π‘‘πœƒγ€— = βˆ’2∫1β–’γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) (sinβ‘πœƒ/cosβ‘πœƒ )Γ—sin⁑〖2πœƒ π‘‘πœƒγ€— = βˆ’2∫1β–’γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) (π‘‘π‘Žπ‘›β‘πœƒ )Γ—sin⁑〖2πœƒ π‘‘πœƒγ€— = βˆ’ 2 ∫1β–’πœƒ sin⁑〖2πœƒ π‘‘πœƒγ€— =βˆ’2(πœƒβˆ«1β–’γ€–sin⁑2πœƒ π‘‘πœƒβˆ’(∫1▒𝑑(πœƒ)/π‘‘πœƒ ∫1β–’sin⁑2πœƒ π‘‘πœƒ) π‘‘πœƒγ€—) =βˆ’2(πœƒ((βˆ’cos⁑2πœƒ)/2)βˆ’βˆ«1β–’1((βˆ’cos⁑2πœƒ)/2) π‘‘πœƒ) =βˆ’2(βˆ’πœƒ(cos⁑2πœƒ/2)•+∫1β–’cos⁑2πœƒ/2 π‘‘πœƒ) =βˆ’2(βˆ’(πœƒ cos⁑2πœƒ)/2•+sin⁑2πœƒ/4) 1/2 γ€–π‘π‘œπ‘ γ€—^(βˆ’1) (π‘₯)=πœƒ πœƒ = 1/2 γ€–π‘π‘œπ‘ γ€—^(βˆ’1) π‘₯ π‘₯^2=γ€–π‘π‘œπ‘ γ€—^2 2πœƒ π‘₯^2=1βˆ’γ€–π‘ π‘–π‘›γ€—^2 2πœƒ 〖𝑠𝑖𝑛〗^2 2πœƒ="1 βˆ’ " π‘₯^2 sin 2πœƒ = √(1βˆ’π‘₯^2 ) Now, x = cos 2πœƒ Putting the values = βˆ’2 (βˆ’1/2 (1/2 γ€–π‘π‘œπ‘ γ€—^(βˆ’1) π‘₯)π‘₯+√(1 βˆ’ π‘₯^2 )/4) = βˆ’2 (√(1 βˆ’ π‘₯^2 )/4βˆ’(π‘₯ γ€–π‘π‘œπ‘ γ€—^(βˆ’1) π‘₯)/4)+ C = 𝟏/𝟐 (𝒙〖 𝒄𝒐𝒔〗^(βˆ’πŸ) π’™βˆ’βˆš(πŸβˆ’π’™^𝟐 ) )+ C

Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh is an IIT Kanpur graduate and has been teaching for 16+ years. At Teachoo, he breaks down Maths, Science and Computer Science into simple steps so students understand concepts deeply and score with confidence.

Many students prefer Teachoo Black for a smooth, ad-free learning experience.