Integration by parts

Chapter 7 Class 12 Integrals
Concept wise

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class

### Transcript

Ex 7.6, 14 γπ₯(logβ‘π₯)γ^2 β«1βγπ₯(logβ‘π₯ )^2.ππ₯ " " γ β΄ β«1βγπ₯(logβ‘π₯ )^2.ππ₯γ=β«1βγ(logβ‘π₯ )^2 π₯ .ππ₯γ = (logβ‘π₯ )^2 β«1βγπ₯ .γ ππ₯ββ«1β((π(logβ‘π₯ )^2)/ππ₯ β«1βγπ₯ .ππ₯γ) ππ₯ = (logβ‘π₯ )^2 . π₯^2/2ββ«1β(2(logβ‘π₯ ) 1/π₯ β«1βγπ₯ .ππ₯γ) ππ₯ Now we know that β«1βγπ(π₯) πβ‘(π₯) γ ππ₯=π(π₯) β«1βπ(π₯) ππ₯ββ«1β(πβ²(π₯)β«1βπ(π₯) ππ₯) ππ₯ Putting f(x) = x and g(x) = (log x)2 = π₯^2/2 (logβ‘π₯ )^2β2β«1βγlogβ‘π₯/π₯ . π₯^2/2γ ππ₯ = π₯^2/2 (logβ‘π₯ )^2ββ«1βγπ₯ logβ‘π₯ γ ππ₯ Solving I1 I1 = β«1βγπ₯ logβ‘π₯ γ ππ₯ β«1βγπ₯ logβ‘π₯ γ ππ₯=β«1β(logβ‘π₯ )π₯ ππ₯ =logβ‘π₯ β«1βπ₯ ππ₯ββ«1β(π(logβ‘π₯ )/ππ₯ β«1βγπ₯.ππ₯γ)ππ₯ Now we know that β«1βγπ(π₯) πβ‘(π₯) γ ππ₯=π(π₯) β«1βπ(π₯) ππ₯ββ«1β(πβ²(π₯)β«1βπ(π₯) ππ₯) ππ₯ Putting f(x) = x and g(x) = log x =logβ‘π₯ (π₯^2/2)ββ«1βγ1/π₯ . π₯^2/2. ππ₯γ =γπ₯^2/2 logγβ‘γ π₯γβ1/2 β«1βγπ₯. ππ₯γ =γπ₯^2/2 logγβ‘π₯β1/2 . π₯^2/2 +πΆ =γπ₯^2/2 πππγβ‘γ π₯γβ π₯^2/4 +πΆ Putting value of I1 in (1), β«1βγπ₯(logβ‘π₯ )^2.ππ₯γ=π₯^2/2 (logβ‘π₯ )^2ββ«1βγ π .πππβ‘π ππγ =π₯^2/2 (logβ‘π₯ )^2β((π₯^2 (logβ‘π₯ ))/2 β π₯^2/4 +πΆ1) =π₯^2/2 (logβ‘π₯ )^2β (π₯^2 (logβ‘π₯ ))/2 + π₯^2/4 βπΆ1 =π^π/π (πππβ‘π )^πβ (π^π (πππβ‘π ))/π + π^π/π+πͺ " "