### Find ∫ secβ‘ x /(1 + cosec x) dx

This is a question of CBSE Sample Paper - Class 12 - 2017/18.

You can download the question paper here https://www.teachoo.com/cbse/sample-papers/

Last updated at Sept. 14, 2018 by Teachoo

This is a question of CBSE Sample Paper - Class 12 - 2017/18.

You can download the question paper here https://www.teachoo.com/cbse/sample-papers/

Transcript

Question 18 Find β«1βsecβ‘π₯/(1 + πππ ππ π₯) dx β«1β(π ππ π₯)/(1 + πππ ππ π₯) ππ₯ = β«1β(1/πππ β‘π₯ )/(1 + 1/π ππβ‘π₯ ) ππ₯ = β«1β(1/πππ β‘π₯ )/((π ππβ‘π₯ + 1)/π ππβ‘π₯ ) ππ₯ = β«1βγ1/πππ β‘π₯ Γ sinβ‘π₯/(π ππβ‘π₯ + 1)γ ππ₯ = β«1β(π ππ π₯)/(πππ π₯(1 + π ππ π₯)) ππ₯ Multiplying and dividing by cos x = β«1β(π ππ π₯)/(πππ π₯(1 + π ππ π₯)) Γcosβ‘π₯/cosβ‘π₯ ππ₯ = β«1β(π ππ π₯ cosβ‘π₯)/(cos^2β‘π₯ (1 + π ππ π₯)) ππ₯ Using cos2 x = 1 β sin2 x = β«1β(π ππ π₯ cosβ‘π₯)/((1 β sin^2β‘π₯ )(1 + sinβ‘γπ₯)γ ) ππ₯ = β«1β(π ππ π₯ cosβ‘π₯)/( (1 β sinβ‘γπ₯)γ (1 + sinβ‘γπ₯)γ (1 + sinβ‘γπ₯)γ ) ππ₯ = β«1β(π ππ π₯ cosβ‘π₯)/((1 + γsinβ‘γπ₯)γγ^2 (1 β sinβ‘γπ₯)γ ) ππ₯ Let sin x = t β΄ cos x dx = dt Putting values in equation = β«1β(π‘ )/((1 + π‘)^2 (1 β π‘)) dt We solve this by partial fractions We can write the integrand as π‘/((1 + π‘)^2 (1 β π‘)) = π΄/(1 + π‘) + π΅/(1 + π‘)^2 + πΆ/(1 β π‘) π‘/((1 + π‘)^2 (1 β π‘)) = (π΄(1 + π‘)(1 β π‘) + π΅(1 β π‘) + πΆ(1 + π‘)^2)/((1 + π‘)^2 (1 β π‘)) By cancelling denominator π‘ = π΄(1 + π‘)(1 β π‘) + π΅(1 β π‘) + πΆ(1 + π‘)^2 Putting t = β1 in (1) β1 = π΄(1+(β1))(1 β(β1)) + π΅(1 β(β1)) + πΆ(1 +(β1))^2 β1 = AΓ0+BΓ2+C(0)^2 β1 = 2B 2B = β1 B = (β1)/2 Putting t = 1 in (1) 1 = π΄(1+1)(1 β1) + π΅(1 β1) + πΆ(1 +1)^2 1 = AΓ0+BΓ0+C(2)^2 1 = 4C 4C = 1 C = 1/4 Putting t = 0 in (1) 0 = π΄(1+0)(1 β0) + π΅(1 β0) + πΆ(1 +0)^2 0 = π΄(1)(1) + π΅(1) + πΆ(1)^2 0 = A+B+C 0 = A + ((β1)/2) + 1/4 1/2β1/4 = A 1/4 = A A = 1/4 Hence we can write β«1β(π‘ )/((1 + π‘)^2 (1 β π‘)) dt = β«1β(1/4)/((1 + π‘) ) dt + β«1β((β1)/2)/(1 + π‘)^2 dt + β«1β(1/4)/( (1 β π‘)) dt

CBSE Class 12 Sample Paper for 2018 Boards

Paper Summary

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18 You are here

Question 19

Question 20

Question 21

Question 22

Question 23

Question 24

Question 25

Question 26

Question 27

Question 28

Question 29

Class 12

Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.