Examples

Chapter 7 Class 12 Integrals
Serial order wise

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### Transcript

Example 7 Find (i) β«1βcos^2β‘π₯ ππ₯ β«1βcos^2β‘π₯ ππ₯ =β«1β((cosβ‘2π₯ + 1)/2) ππ₯ = 1/2 β«1β(cosβ‘2π₯+1) ππ₯ = 1/2 [β«1βcosβ‘2π₯ ππ₯+β«1β1 ππ₯] We know cosβ‘2π₯=2 cos^2β‘π₯β1 cosβ‘2π₯+1=2 cos^2β‘π₯ (cosβ‘2π₯ + 1)/2=cos^2β‘π₯ β«1βπππβ‘ππ ππ Let 2π₯=π‘ 2 =ππ‘/ππ₯ ππ₯=1/2 ππ‘ β«1βcosβ‘π‘ . 1/2 ππ‘ =1/2 (sinβ‘π‘+πΆ1) Putting value of π‘ = 2π₯ =1/2 sinβ‘2π₯+πΆ1 β«1βπ ππ =β«1βπ₯^0 ππ₯ =[π₯^(0 + 1)/(0 + 1)]+πΆ =π₯+πΆ2 Thus, β«1βcos^2β‘π₯ ππ₯=1/2 [β«1βcosβ‘2π₯ ππ₯+β«1β1 ππ₯] =1/2 [1/2 sinβ‘2π₯+πΆ1+π₯+πΆ2] =1/4 sinβ‘2π₯+π₯/2+1/2(πΆ1+πΆ2) =π/π+π/π π¬π’π§β‘ππ+πͺ ("From (1) and (2) " ) ("Let" πΆ1+πΆ2=πΆ)