Indefinite integration.pdf
- 1. Revision Session Indefinite Integration ©2021, BYJU'S. All rights reserved
- 2. Sub-Topic Number of Questions Integration by Substitution 21 Integration of Special Functions Integration by Parts 6 Integration by Partial Fraction 2 MAIN ADVANCED 1 0 0 0 PAST 10 YEARS JEE MAIN & JEE ADVANCED ANALYSIS (2012-2021) JEE MAIN : 65 Papers, 41 Questions JEE ADVANCED : 20 Papers, 1 Question 12 ©2021, BYJU'S. All rights reserved
- 3. JEE Main Aug 2021 If Then the value of 𝑎4 + 𝑏5 is ___. If tan 𝑥 + cot 𝑥 𝑑𝑥 = 𝑎 tan−1 tan 𝑥−1 𝑏 tan 𝑥 + 𝑐. ©2021, BYJU'S. All rights reserved
- 4. JEE Main Aug 2021 If න 𝑑𝑥 𝑥2 + 𝑥 + 1 2 = 𝑎 tan−1 2𝑥 + 1 3 + 𝑏 2𝑥 + 1 𝑥2 + 𝑥 + 1 + 𝐶, 𝑥 > 0, where 𝑐 is a constant of integration, then the value of 9( 3𝑎 + 𝑏) is equal to ____. ©2021, BYJU'S. All rights reserved
- 5. JEE Main Jul 2021 න 𝑒𝑥 2 − 𝑥2 1 − 𝑥 1 − 𝑥2 𝑑𝑥 Evaluate a 𝑒𝑥 1+𝑥 1−2𝑥 + 𝑐 b 𝑒𝑥 1+𝑥 1−𝑥 + 𝑐 c 𝑒𝑥 1−𝑥 1+𝑥 + 𝑐 d 𝑒𝑥 1+𝑥 1−𝑥 2 + 𝑐 ©2021, BYJU'S. All rights reserved
- 6. JEE Main Mar 2021 The integral is equal to ____. (where 𝐶 is a constant of integration) a 1 2 sin 2𝑥 − 1 2 + 5 + 𝐶 b 1 2 cos 2𝑥 + 1 2 + 5 + 𝐶 c 1 2 cos 2𝑥 − 1 2 + 5 + 𝐶 d 1 2 sin 2𝑥 + 1 2 + 5 + 𝐶 න 2𝑥 − 1 cos 2𝑥 − 1 2 + 5 4𝑥2 − 4𝑥 + 6 𝑑𝑥 ©2021, BYJU'S. All rights reserved
- 7. a 1 2 sin 2𝑥 − 1 2 + 5 + 𝐶 b 1 2 cos 2𝑥 + 1 2 + 5 + 𝐶 c 1 2 cos 2𝑥 − 1 2 + 5 + 𝐶 d 1 2 sin 2𝑥 + 1 2 + 5 + 𝐶 න 2𝑥 − 1 cos 2𝑥 − 1 2 + 5 4𝑥2 − 4𝑥 + 6 𝑑𝑥 ©2021, BYJU'S. All rights reserved
- 8. JEE Main Aug 2021 න sin−1 2𝑥 + 2 4𝑥2 + 8𝑥 + 13 𝑑𝑥 = 𝐴 tan−1 2𝑥 + 2 3 + 𝐵 ln 4𝑥2 + 8𝑥 + 13 + 𝐶 If a 3 2 𝑥 + 1 , −3 b 3 2 𝑥 + 1 , −6 c 𝑥 + 1, −3 d (𝑥 + 1, −6) (where 𝐶 is a constant of integration), then the ordered pair 𝐴, 4𝐵 is equal to ____. ©2021, BYJU'S. All rights reserved
- 9. න sin−1 2𝑥 + 2 4𝑥2 + 8𝑥 + 13 𝑑𝑥 = 𝐴 tan−1 2𝑥 + 2 3 + 𝐵 ln 4𝑥2 + 8𝑥 + 13 + 𝐶 a 3 2 𝑥 + 1 , −3 b 3 2 𝑥 + 1 , −6 c 𝑥 + 1, −3 d (𝑥 + 1, −6) ©2021, BYJU'S. All rights reserved
- 10. JEE Main Feb 2021 න cos 𝑥 − sin 𝑥 8 − sin 2𝑥 𝑑𝑥 = 𝑎 sin−1 sin 𝑥 + cos 𝑥 𝑏 + 𝐶 a −1, 3 b (3, 1) c 1,3 d 1, −3 (where 𝐶 is a constant of integration), then the ordered pair (𝑎, 𝑏) is equal to ____. ©2021, BYJU'S. All rights reserved
- 11. න cos 𝑥 − sin 𝑥 8 − sin 2𝑥 𝑑𝑥 = 𝑎 sin−1 sin 𝑥 + cos 𝑥 𝑏 + 𝐶 a −1, 3 b (3, 1) c 1,3 d 1, −3 ©2021, BYJU'S. All rights reserved
- 12. JEE Main Feb 2021 න 1 − 5 sin2 𝑥 cos5 𝑥 sin2 𝑥 𝑑𝑥 = 𝑓 𝑥 cos5 𝑥 + 𝐶 Then find the value of 𝑓 3𝜋 4 . . (where 𝐶 is a constant of integration), ©2021, BYJU'S. All rights reserved
- 13. Let 𝑓 𝑥−4 𝑥+2 = 2𝑥 + 1, 𝑥 ∈ ℝ − 1, −2 , then 𝑓 𝑥 𝑑𝑥 is equal to : (where 𝐶 is the constant of integration) JEE Main 2018 a 12 ln 1 − 𝑥 − 3𝑥 + 𝐶 c 12 ln 1 − 𝑥 + 3𝑥 + 𝐶 b −12 ln 1 − 𝑥 − 3𝑥 + 𝐶 d −12 ln 1 − 𝑥 + 3𝑥 + 𝐶 ©2021, BYJU'S. All rights reserved
- 14. 𝑓 𝑥 − 4 𝑥 + 2 = 2𝑥 + 1 a 12 ln 1 − 𝑥 − 3𝑥 + 𝐶 c 12 ln 1 − 𝑥 + 3𝑥 + 𝐶 b −12 ln 1 − 𝑥 − 3𝑥 + 𝐶 d −12 ln 1 − 𝑥 + 3𝑥 + 𝐶 ©2021, BYJU'S. All rights reserved
- 15. If , then is: JEE Main Mar 2021 𝑓 𝑥 = න 5𝑥8 + 7𝑥6 𝑥2 + 1 + 2𝑥7 2 𝑑𝑥, 𝑥 ≥ 0 , 𝑓 0 = 0 and 𝑓 1 = 1 𝑘 the value of 𝑘 ©2021, BYJU'S. All rights reserved
- 16. 𝑒3 ln 2𝑥+5𝑒2 ln 2𝑥 𝑒4 ln 𝑥+5𝑒3 ln 𝑥−7𝑒2 ln 𝑥 𝑑𝑥 If 𝑒3 ln 2𝑥+5𝑒2 ln 2𝑥 𝑒4 ln 𝑥+5𝑒3 ln 𝑥−7𝑒2 ln 𝑥 𝑑𝑥 , 𝑥 > 0, is equal to: (where 𝑐 is integration constant) JEE Main Feb 2021 a b c d ln |𝑥2 + 5𝑥 − 7| + 𝑐 1 4 ln |𝑥2 + 5𝑥 − 7| + 𝑐 4 ln |𝑥2 + 5𝑥 − 7| + 𝑐 ln |𝑥2 + 5𝑥 − 7| + 𝑐 ©2021, BYJU'S. All rights reserved
- 17. 𝑒3 ln 2𝑥+5𝑒2 ln 2𝑥 𝑒4 ln 𝑥+5𝑒3 ln 𝑥−7𝑒2 ln 𝑥 𝑑𝑥 a b c d ln |𝑥2 + 5𝑥 − 7| + 𝑐 1 4 ln |𝑥2 + 5𝑥 − 7| + 𝑐 4 ln |𝑥2 + 5𝑥 − 7| + 𝑐 ln |𝑥2 + 5𝑥 − 7| + 𝑐 Let 𝐼 = ©2021, BYJU'S. All rights reserved
- 18. If 1 4 𝑥−1 3 𝑥+2 5 𝑑𝑥 , 𝑥 > 0, is equal to: (where 𝑐 is integration constant) JEE Main Aug 2021 a 3 4 𝑥+2 𝑥−1 5 4 + 𝑐 c 3 4 𝑥+2 𝑥−1 1 4 + 𝑐 b 4 3 𝑥−1 𝑥+2 5 4 + 𝑐 d 4 3 𝑥−1 𝑥+2 1 4 + 𝑐 1 4 𝑥−1 3 𝑥+2 5 𝑑𝑥 ©2021, BYJU'S. All rights reserved
- 19. 1 4 𝑥−1 3 𝑥+2 5 𝑑𝑥 Let 𝐼 = a 3 4 𝑥+2 𝑥−1 5 4 + 𝑐 c 3 4 𝑥+2 𝑥−1 1 4 + 𝑐 b 4 3 𝑥−1 𝑥+2 5 4 + 𝑐 d 4 3 𝑥−1 𝑥+2 1 4 + 𝑐 ©2021, BYJU'S. All rights reserved
- 20. JEE Main Jan 2020 If cos 𝑥 sin3 𝑥 1+sin6 𝑥 2 3 𝑑𝑥 (where 𝑐 is integration constant) then 𝜆𝑓 𝜋 3 a − 9 8 b 9 8 c 2 d −2 = 𝑓 𝑥 1 + sin6 𝑥 1/𝜆 + 𝑐, cos 𝑥 sin3 𝑥 1+sin6 𝑥 2 3 𝑑𝑥 ©2021, BYJU'S. All rights reserved
- 21. cos 𝑥 sin3 𝑥 1+sin6 𝑥 2 3 𝑑𝑥 a − 9 8 b 9 8 c 2 d −2 ©2021, BYJU'S. All rights reserved
- 22. JEE Main Jan 2020 If 𝑑𝑥 𝑥+4 8 7 𝑥−3 6 7 , 𝑥 > 0, is equal to: (where 𝑐 is integration constant) a − 𝑥−3 𝑥+4 − 1 7 + 𝑐 c 𝑥−3 𝑥+4 1 7 + 𝑐 b 1 2 𝑥−3 𝑥+4 3 7 + 𝑐 d −1 13 𝑥−3 𝑥+4 − 13 7 + 𝑐 𝑑𝑥 𝑥+4 8 7 𝑥−3 6 7 ©2021, BYJU'S. All rights reserved
- 23. 𝑑𝑥 𝑥+4 8 7 𝑥−3 6 7 Let 𝐼 = a − 𝑥−3 𝑥+4 − 1 7 + 𝑐 c 𝑥−3 𝑥+4 1 7 + 𝑐 b 1 2 𝑥−3 𝑥+4 3 7 + 𝑐 d −1 13 𝑥−3 𝑥+4 − 13 7 + 𝑐 ©2021, BYJU'S. All rights reserved
- 24. JEE Main Jan 2020 If 𝑑𝜃 cos2 𝜃 tan 2𝜃+sec 2𝜃 = 𝜆 tan 𝜃 + 2 ln 𝑓 𝜃 + 𝑐, (where 𝑐 is integration constant) then the ordered pair 𝜆, 𝑓 𝜃 is equal to: a −1, 1 − tan 𝜃 b −1, 1 + tan 𝜃 c 1, 1 + tan 𝜃 d −1, 1 − tan 𝜃 𝑑𝜃 cos2 𝜃 tan 2𝜃+sec 2𝜃 = 𝜆 tan 𝜃 + 2 ln 𝑓 𝜃 + 𝑐 ©2021, BYJU'S. All rights reserved
- 25. 𝑑𝜃 cos2 𝜃 tan 2𝜃+sec 2𝜃 = 𝜆 tan 𝜃 + 2 ln 𝑓 𝜃 + 𝑐 a −1, 1 − tan 𝜃 b −1, 1 + tan 𝜃 c 1, 1 + tan 𝜃 d −1, 1 − tan 𝜃 ©2021, BYJU'S. All rights reserved
- 26. JEE Main Apr 2019 a 𝑥 − 𝛼 and ln sin 𝑥 − 𝛼 Let 𝛼 ∈ 0, 𝜋 2 be fixed. If the tan 𝑥+tan 𝛼 tan 𝑥−tan 𝛼 𝑑𝑥 = 𝐴 𝑥 cos 2𝛼 + 𝐵 𝑥 sin 2𝛼 + 𝑐, (where 𝑐 is integration constant) then the functions 𝐴(𝑥) and 𝐵(𝑥) are respectively: b 𝑥 + 𝛼 and ln sin 𝑥 − 𝛼 c 𝑥 − 𝛼 and ln cos 𝑥 − 𝛼 d 𝑥 + 𝛼 and ln sin 𝑥 + 𝛼 ©2021, BYJU'S. All rights reserved
- 27. tan 𝑥+tan 𝛼 tan 𝑥−tan 𝛼 𝑑𝑥 = 𝐴 𝑥 cos 2𝛼 + 𝐵 𝑥 sin 2𝛼 + 𝑐 a 𝑥 − 𝛼 and ln sin 𝑥 − 𝛼 b 𝑥 + 𝛼 and ln sin 𝑥 − 𝛼 c 𝑥 − 𝛼 and ln cos 𝑥 − 𝛼 d 𝑥 + 𝛼 and ln sin 𝑥 + 𝛼 a 𝑥 − 𝛼 and ln sin 𝑥 − 𝛼 b 𝑥 + 𝛼 and ln sin 𝑥 − 𝛼 ©2021, BYJU'S. All rights reserved