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Integration (Calculus)

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Published in: Mathematics
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Integration for class 12th. It covers all type of questions.

Rajendra K / Bangalore

More than 25 years of teaching experience

Qualification: M.Phil (punjab university chandigarh - 1977)

Teaches: Mathematics, AIEEE, IIT JEE Advanced, IIT JEE Mains, WBJEE

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  1. Notes, Formula, Points and some Techniques Basic Formula Group A 1. 2. 3. 4. 5. 6. Group B 7. 8. 9. 10. 11. 12. sinxdx = — cos x + C cos xdx = sin x + C sec xdx=tanx + C co sec xdx= —cot x sec x tan xdx = sec x co sec x cot xdx = —co sec x 11-1 xndx = e dx = a n 4-1 x a — cos(ax+ b) sin(ax + b)dx = a sin(ax + b) f cos(ax + b)dx = a tan(ax + b) see (ax + b)dx = a — cot(ax + b) cos ec2 (ax + b)dx = a sec(ax + b) sec x tan x(ax + b)dx = a —co sec(ax + b) co sec x cot x(ax + b)dx = a (ax + b)n+l f (ax + bYdx— (ax+b) (ax+b) a (bx+c) 1 a (bx+c) loge a • b 1 log(ax + b) loge a —dx = log x + C 1 -1 —dx = fldx = x + C dx = (ax + b) 1 dx = a -1 n — 1) (ax + (ax + 1 a Group C 13. 14. 15. 16. tan xdx = log sec x + C cot xdx = log sin x + C sec xdx = log(sec x + tan x) + C log sec(ax + b) tan(ax + b)dx — a log sin(ax + b) f cot(ax + b)dx = a log(sec(ax + b) + tan(ax + b)) sec(ax + b)dx — a log(co sec(ax + b) — cot(ax + b)) co secxdx = log(co sec x— cot x) +C co sec(ax + b)dx = a
  2. 2 Group D ex = t 17. 18 19 1 sin x + C 1 sec- x + C x x2—1 1 dx=tan x + C 2 Some Other Points 20. 21. 22. 23. 24. sin tan sec sin tan sec sin X, COS X cot x X, X, cosec x X, COS X x, cot x + X, cosec x X, COS X use half angle formula use sec x -1 and cosec x -1 direct split one odd power split two odd power split two odd power and use by parts use half angle formula twice Use following substitution a. b. c. d. f. TX = t x = t 2 dx=2tdt log x tan x = t x=et dx=etdt dt = t x = tan I t dt = dt CIX = dt a Inverse function = t if (X) [f/ (x) dx], put f (x) = t f/ (x) dx=dt There are some pair of f (x) and . f/ (x) may be with out constant. Some important pairs f(x) xe f/ (x) may be with out constant (x -k I)ex
  3. 3 1 4 x log(sec x + tan x) 2 cos x 25. Remove under roots by following substitution x—a x—a 26. 27. a. b. d. f. I—sin x 1 + tan sec = cos x x = sec 0 — tan 0 2 put x put x put x put x = cos 20 or x or (x—a —x) (x—a )(x dx 1 5 x 1 2 sec x sin 2x = 3 sin 0 2 = 3 tan20 2 =3sec2 0 — acos2 0+ ßsin2 0 — asec2 0 — p tan 2 0 Deg(Nr) Deg (Nr) a. b. sin x(terms sin x or cos x) dx cos x(terms cos x or sin x) put x — put x — divide by Dr. in Nr. divide and multiply by sin x divide and multiply by cos x

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