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Integrate sin^4 x  - Teachoo Maths - Chapter 7 Class 12 - Ex 7.3

Ex 7.3, 10 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.3, 10 - Chapter 7 Class 12 Integrals - Part 3

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Ex 7.3, 10 Integrate the function 𝑠𝑖𝑛4 π‘₯ ∫1β–’sin^4⁑π‘₯ 𝑑π‘₯ =∫1β–’(sin^2⁑π‘₯ )^2 𝑑π‘₯ =∫1β–’((1 βˆ’ cos⁑2π‘₯)/2)^2 𝑑π‘₯ =1/4 ∫1β–’(1βˆ’cos⁑2π‘₯ )^2 𝑑π‘₯ We know that π‘π‘œπ‘ β‘2πœƒ=1βˆ’2 〖𝑠𝑖𝑛〗^2β‘πœƒ 2 〖𝑠𝑖𝑛〗^2β‘πœƒ=1βˆ’π‘π‘œπ‘ β‘2πœƒ 〖𝑠𝑖𝑛〗^2β‘πœƒ=(1 βˆ’ π‘π‘œπ‘ β‘2πœƒ)/2 Replace πœƒ by π‘₯ 〖𝑠𝑖𝑛〗^2⁑π‘₯=(1 βˆ’ π‘π‘œπ‘ β‘2π‘₯)/2 =1/4 ∫1β–’(1^2+(cos⁑2π‘₯ )^2βˆ’2(1)(cos⁑2π‘₯ )) 𝑑π‘₯ =1/4 ∫1β–’(1+cos^2⁑2π‘₯βˆ’2 cos⁑2π‘₯ ) 𝑑π‘₯ =1/4 ∫1β–’(1+(1 + cos⁑4π‘₯)/2βˆ’2 cos⁑2π‘₯ ) 𝑑π‘₯ =1/4 ∫1β–’1 𝑑π‘₯+1/8 ∫1β–’(1+cos⁑4π‘₯ ) 𝑑π‘₯βˆ’2/4 ∫1β–’cos⁑2π‘₯ 𝑑π‘₯ =1/4 ∫1β–’1 𝑑π‘₯+1/8 ∫1β–’1 𝑑π‘₯+1/8 ∫1β–’cos⁑4π‘₯ 𝑑π‘₯βˆ’1/2 ∫1β–’cos⁑2π‘₯ 𝑑π‘₯ We know that cos⁑2πœƒ=2 cos^2β‘γ€–πœƒβˆ’1γ€— cos⁑〖2πœƒ+1γ€—=2 cos^2β‘πœƒ Replace πœƒ by 2π‘₯ cos⁑〖4π‘₯+1γ€—=2 cos^2⁑2π‘₯ cos⁑〖4π‘₯ + 1γ€—/2=cos^2⁑2π‘₯ ∫1β–’cos⁑(π‘Žπ‘₯+𝑏) 𝑑π‘₯=𝑠𝑖𝑛⁑(π‘Žπ‘₯ + 𝑏)/π‘Ž =π‘₯/4 + π‘₯/8 + 1/8 sin⁑4π‘₯/4 βˆ’ 1/2 sin⁑2π‘₯/2 +𝐢 =(2π‘₯ + π‘₯)/8 + 1/32 sin⁑4π‘₯βˆ’ 1/4 sin⁑2π‘₯+𝐢 =πŸ‘π’™/πŸ–βˆ’ 𝟏/πŸ’ π’”π’Šπ’β‘πŸπ’™++ 𝟏/πŸ‘πŸ π’”π’Šπ’β‘πŸ’π’™+π‘ͺ

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