Example 25 (iv) - Chapter 7 Class 12 Integrals
Last updated at April 16, 2024 by Teachoo
Examples
Example 1 (i)
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Chapter 7 Class 12 Integrals
Serial order wise
Transcript
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
