Integration by specific formulaes - Formula 3
Integration by specific formulaes - Formula 3
Last updated at April 16, 2024 by Teachoo
Ex 7.4, 1 (3π₯^2)/(π₯^6 + 1) We need to find β«1β(ππ^π)/(π^π + π) π π Let π^π=π Diff both sides w.r.t. x 3π₯^2=ππ‘/ππ₯ π π=π π/(ππ^π ) Thus, our equation becomes β«1β(ππ^π)/(π^π + π) π π =β«1β(3π₯^2)/((π₯^3 )^2 + 1) ππ₯ Putting the value of π₯^3=π‘ and ππ₯=ππ‘/(3π₯^2 ) =β«1β(3π₯^2)/(π‘^2 + 1) .ππ‘/(3π₯^2 ) =β«1βππ‘/(π‘^2 + 1) =β«1βπ π/(π^π + (π)^π ) =1/1 tan^(β1)β‘γ π‘/1 γ+πΆ It is of form β«1βππ‘/(π₯^2 + π^2 ) =1/π γγπ‘ππγ^(β1) γβ‘γπ₯/πγ +πΆ β΄ Replacing π₯ = π‘ and π by 1 , we get =tan^(β1)β‘γ (π‘)γ+πΆ =γγπππγ^(βπ) γβ‘(π^π )+πͺ ("Using" π‘=π₯^3 )