1. Chapter 7 Class 12 Integrals (Term 2)
2. Concept wise
3. Integration by substitution - Trignometric - Normal

Transcript

Ex 7.2, 32 Integrate 1/(1 + cotβ‘π₯ ) Simplify the given function β«1β1/(1 + cotβ‘π₯ ) ππ₯ = β«1β1/(1 + cosβ‘π₯/sinβ‘π₯ ) ππ₯ = β«1β1/(γsinβ‘π₯ + cosγβ‘π₯/sinβ‘π₯ ) ππ₯ = β«1βsinβ‘π₯/γsinβ‘π₯ + cosγβ‘π₯ ππ₯ Multiplying & dividing by 2 = β«1β(2 sinβ‘π₯)/2(γsinβ‘π₯ + cosγβ‘π₯ ) ππ₯ Adding & subtracting πππ β‘π₯ in numerator = β«1β(sinβ‘π₯ + sinβ‘π₯ + cosβ‘π₯ β cosβ‘π₯)/2(γsinβ‘π₯ + cosγβ‘π₯ ) ππ₯ = 1/2 β«1β((sinβ‘π₯ + cosβ‘π₯ + sinβ‘π₯ β cosβ‘π₯)/γsinβ‘π₯ + cosγβ‘π₯ ) ππ₯ = 1/2 β«1β((sinβ‘π₯ + cosβ‘π₯)/γsinβ‘π₯ + cosγβ‘π₯ +(sinβ‘π₯ β cosβ‘π₯)/γsinβ‘π₯ + cosγβ‘π₯ ) ππ₯ = 1/2 β«1β(1+(sinβ‘π₯ β cosβ‘π₯)/γsinβ‘π₯ + cosγβ‘π₯ ) ππ₯ = 1/2 [π₯+β«1β((sinβ‘π₯ β cosβ‘π₯)/γsinβ‘π₯ + cosγβ‘π₯ ) ππ₯] + πΆ1 β¦(1) Solving π1 I1 = β«1β(sinβ‘π₯ β cosβ‘π₯)/γsinβ‘π₯ + cosγβ‘π₯ ππ₯ Let γsinβ‘π₯ + cosγβ‘π₯=π‘ Differentiating both sides π€.π.π‘.π₯ γcosβ‘π₯βsinγβ‘π₯=ππ‘/ππ₯ ππ₯=ππ‘/γcosβ‘π₯ β sinγβ‘π₯ ππ₯=ππ‘/(β(γsinβ‘π₯ β cosγβ‘π₯ ) ) Thus, our equation becomes β¦(2) I1 = β«1β(sinβ‘π₯ β cosβ‘π₯)/γsinβ‘π₯ + cosγβ‘π₯ ππ₯ = β«1β(sinβ‘π₯ β cosβ‘π₯)/π‘ . ππ‘/(β(γsinβ‘π₯ β cosγβ‘π₯ ) ) = β1β«1βππ‘/π‘ = βγlog γβ‘|π‘|+πΆ Putting back π‘=π ππβ‘π₯+πππ β‘π₯ = βlogβ‘γ |sinβ‘π₯+cosβ‘π₯ |γ+πΆ2 Putting the value of I1 in (1) β΄ β«1βγ1/(1 + cotβ‘π₯ ) " " γ = 1/2 [π₯+β«1β((sinβ‘π₯ β cosβ‘π₯)/γsinβ‘π₯ + cosγβ‘π₯ ) ππ₯] + πΆ1 = 1/2 [π₯βlogβ‘|sinβ‘π₯+cosβ‘π₯ |+πΆ2" " ] +πΆ1 = π₯/2β1/2 logβ‘γ |sinβ‘π₯+cosβ‘π₯ |γ+πΆ1+πΆ2/2 = π/π βπ/π πππβ‘γ |πππβ‘π+πππβ‘π |γ+πͺ

Integration by substitution - Trignometric - Normal