Example 25 (iv) - Chapter 7 Class 12 Integrals
Last updated at Dec. 16, 2024 by Teachoo
Last updated at Dec. 16, 2024 by Teachoo
Example 25 Evaluate the following integrals: (iv) β«_0^(π/4)βγsin^3β‘2π‘ cosβ‘2 π‘γ ππ‘ Let F(π₯)=β«1βγπ ππ^3 2π‘ πππ 2π‘ ππ‘γ Let sππ 2π‘=π’ Differentiating w.r.t.π₯ (π(sinβ‘2π‘))/ππ‘=ππ’/ππ‘ 2cππ 2π‘ =ππ’/ππ‘ ππ‘=ππ’/(2 πππ 2π‘) Putting value of u and du in our integral β«1βγπ ππ^3 2π‘ πππ 2π‘ ππ‘γ=β«1βγπ’^3 πππ 2π‘ Γ ππ’/(2 πππ 2π‘)γ =1/2 β«1βγπ’^3 ππ’γ =1/2 π’^(3+1)/(3+1)=1/2 π’^4/4= π’^4/8 Putting back π’=π ππ 2π‘ =1/8 π ππ^4 2π‘ Hence, F(π‘)=1/8 π ππ^4 2π‘ Now, β«_0^(π/4)βγπ ππ^3 2π‘ πππ 2π‘=πΉ(π/4)βπΉ(0) γ =1/8 π ππ^4 2(π/4)β1/8 π ππ^4 2(0) =1/8 π ππ^4 π/2β1/8 π ππ^4 (0) =1/8 Γ1^4β1/8 Γ0^4 =1/8 Γ1β0 =π/π
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About the Author
Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo