Find f log x (1 + log x) 2 dx
This question is similar to Ex 7.2, 35 - Chapter 7 Class 12 - Integrals


CBSE Class 12 Sample Paper for 2022 Boards (For Term 2)
CBSE Class 12 Sample Paper for 2022 Boards (For Term 2)
Last updated at Dec. 14, 2024 by Teachoo
This question is similar to Ex 7.2, 35 - Chapter 7 Class 12 - Integrals
Transcript
Question 1 β Choice 1 Find β«1βγlogβ‘π₯/(1 + logβ‘π₯ )^2 ππ₯γLet π=β«1βγlogβ‘π₯/(1 + logβ‘π₯ )^2 ππ₯γ =β«1βγ(πππβ‘π + π β π)/(1 + logβ‘π₯ )^2 ππ₯γ =β«1βγ((1 + logβ‘π₯) β 1)/(1 + logβ‘π₯ )^2 ππ₯γ =β«1βγ((1 + logβ‘π₯) )/(1 + logβ‘π₯ )^2 ππ₯γββ«1βγ1/(1 + logβ‘π₯ )^2 ππ₯γ =β«1βγ(π )/((π + πππβ‘π ) ) π πγββ«1βγπ/(π + πππβ‘π )^π π πγ Solving β«1βγ(π )/((π + π₯π¨π β‘π± ) ) ππ±γ Using Integration by parts β«1βγ(π )/((π + πππβ‘π ) ) π πγ = β«1βγ(1 )/((1 + πππβ‘π₯ ) ) Γ 1 ππ₯γ = 1/((1 + logβ‘π₯)) β«1βγ1 ππ₯γββ«1β(π (π/(π + πππ π))/π π β«1βγπ π πγ) π π = 1/((1 + logβ‘π₯))Γ π₯ββ«1β((βπ)/(π +πππ π)^π Γπ/π Γ π) π π = π₯/((1 + logβ‘π₯))+β«1βπ/(π + πππ π)^π π πWe know that β«1βγπ(π₯) πβ‘(π₯) γ ππ₯=π(π₯) β«1βπ(π₯) ππ₯ββ«1β(π^β² (π₯) β«1βπ(π₯) ππ₯) ππ₯ Putting f(x) = 1/(1 + log x) and g(x) = 1 Thus I=β«1βγ(1 )/((1 + πππβ‘π₯ ) ) ππ₯γββ«1βγ1/(1 + πππβ‘π₯ )^2 ππ₯γ = π₯/((1 + logβ‘π₯))+β«1β1/(1 + πππ π₯)^2 ππ₯ββ«1βγ1/(1 + πππβ‘π₯ )^2 ππ₯γ = π/((π + πππβ‘π))+πͺ