Examples

Chapter 7 Class 12 Integrals
Serial order wise

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Example 27 Evaluate the following integrals: (iv) β«_0^(π/4)βγsin^3β‘2π‘ cosβ‘2 π‘γ ππ‘ Let F(π₯)=β«1βγπ ππ^3 2π‘ πππ  2π‘ ππ‘γ Let sππ 2π‘=π’ Differentiating w.r.t.π₯ (π(sinβ‘2π‘))/ππ‘=ππ’/ππ‘ 2cππ  2π‘ =ππ’/ππ‘ ππ‘=ππ’/(2 πππ  2π‘) Putting value of u and du in our integral β«1βγπ ππ^3 2π‘ πππ  2π‘ ππ‘γ=β«1βγπ’^3 πππ  2π‘ Γ ππ’/(2 πππ  2π‘)γ =1/2 β«1βγπ’^3 ππ’γ =1/2 π’^(3+1)/(3+1)=1/2 π’^4/4= π’^4/8 Putting back π’=π ππ 2π‘ =1/8 π ππ^4 2π‘ Hence, F(π‘)=1/8 π ππ^4 2π‘ Now, β«_0^(π/4)βγπ ππ^3 2π‘ πππ  2π‘=πΉ(π/4)βπΉ(0) γ =1/8 π ππ^4 2(π/4)β1/8 π ππ^4 2(0) =1/8 π ππ^4 π/2β1/8 π ππ^4 (0) =1/8 Γ1^4β1/8 Γ0^4 =1/8 Γ1β0 =π/π