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Class XII Mathematics Mathematics Part-II 1 +1 EXERCISE7.6 Integrate the functions in Exercises 1 to 22. 1. x sin x 2. x sin 3x 3. x e x 4. x log x 5. x log 2x 6. x log x 7. x sin x 1 8. x ta...
Class XII Mathematics Mathematics Part-II 1 2 2 x − a + a = x x − a − ∫ 2 2 dx = x x −a − ∫ 2 2 dx x −a x − a Reprint 2026-27 INTEGRALS 265 dx = x x −a − 2 x −a dx−a 2 ∫ ∫ x −a 2 2 2 2 dx = x...
Class XII Mathematics Mathematics Part-II 1 x + 1 = y so that dx = dy. 2 2 Thus ∫ 3 − x− x dx = ∫ 4− y dy 1 y 4− y +2 4 sin–1y +C = 2 2 2 [using7.6.2(iii)] 1 2 –1x +1  = 2(x+1) 3−2 − x x + ...
Class XII Mathematics Mathematics Part-II 1 called the upper limit of the integral. The definite integral is introduced either as the limit of a sum or if it has an anti derivative F in the i...
Class XII Mathematics Mathematics Part-II 1 on the closed interval [a,b] and F be b an anti derivative of f. Then ∫ f (x)dx = [F( )] a F (b) – F(a). a Remarks (i) Inwords,theTheorem2tellsusth...
Class XII Mathematics Mathematics Part-II 1 Since ∫ x dx = = F(x) , Therefore, by the second fundamental theorem, we get I = F(3) – F(2) = 27 – 8 =9 3 3 3 9 x (ii) Let I =∫ 4 3 dx . We first ...
Class XII Mathematics Mathematics Part-II 1 Put sin 2t = u so that 2 cos 2t dt = du or cos 2t dt = du So sin 2t cos2t dt = 1 u du ∫ 2 ∫ 1 4 1 4 = [u ]= sin 2t = F(t)say 8 8 Therefore, by the ...
Class XII Mathematics Mathematics Part-II 1 previous sections, we have discussed several methods for finding the indefinite integral.Oneoftheimportantmethodsforfindingtheindefiniteintegralist...
Class XII Mathematics Mathematics Part-II 1 4 5 2 Therefore ∫ −15x x +1 dx = ∫0 t dt   2  3 3 = 2   = 2 2 –0 2 = 2 (2 2) = 4 2 3   3   3 3 1 tan–1 x Example 27 Evaluate ∫ 2 dx 0 1...
Class XII Physics Physics Part-II 4 most of an atom is empty space. With the atom being largely empty space, it is easy to see why most α-particles go right through a thin metal foil....
Class XII Mathematics Mathematics Part-II 1 sinx (C) x cosx (D) sinx + x cosx 7.10 Some Properties of Definite Integrals We list below some important properties of definite integrals. These w...
Class XII Mathematics Mathematics Part-II 1 ( )x = 0 . a Proof of P Let F be anti derivative of f. Then b ∫a f (x) dx = F(b) – F(a) ... (1) c ∫ f (x) dx = F(c) – F(a) ... (2) a b and ∫c f (x)...
Class XII Mathematics Mathematics Part-II 1 Hence ∫ 0 f (x) dx = ∫0 f (x) dx + ∫0 f (2a− x) dx 2af (x) dx = a f (x) dx + af (2a − x) dx Proof of P Us6ng P , we5have ∫ 0 ∫0 ∫ 0 ... (1) Now,if ...
Class XII Mathematics Mathematics Part-II 1 Solution We note that x – x ≥ 0 on [– 1, 0] and x – x ≤ 0 on [0, 1] and that x – x ≥ 0 on [1, 2]. So by P 2e write 2 0 1 2 ∫ x – x dx = ∫ (x – x) d...
Class XII Mathematics Mathematics Part-II 1 x π xsin x Solution Let I = ∫ 01+cos x 2 dx . Then, by P ,4we have π I = (π− x)sin ( π− x) dx ∫ 0 1 cos (2 π− x) π (π− x)sin x dx π sin x dx = ∫ 2 ...
Class XII Mathematics Mathematics Part-II 1 4 2 2 dx 2 cos x I = ∫0 4 π 4 π = ∫ 0 cos4 x + si x dx ... (2) sin ( − x +cos ( − x) 2 2 Adding (1) and (2), we get π sin4 x+ cos 4x π π π 2I = ∫ 2...
Class XII Mathematics Mathematics Part-II 1 t in the first integral. Then 2 dx = dt, when x = 0, t = 0 and whex = , t = π. Therefore 2I = 1 πlog sint dt − π log2 2 ∫ 0 2 π = 2 2 logsint dt − ...
Class XII Mathematics Mathematics Part-II 1 20. The value of ∫−π (x + x cosx + tan x +1) dx is (A) 0 (B) 2 (C) π (D) 1 π 2  4+3sin x  21. The value of ∫0 log   dx is 4 +cos x (A) 2 (B) ...
Class XII Mathematics Mathematics Part-II 1 1, 1 1 which give A = , B= C= – . Substituting values ofA, B and C in (2), we get 2 2 1 = 1 − 1 x − 1 ... (3) (x −1) (x +1) 2(x −1) 2 (x2 +1) 2(x2 ...
Class XII Mathematics Mathematics Part-II 1   ∫ Put tan x = t , so that sec x dx = 2t dt or dx = 2t dt 1+t 4  1  2t Then I = ∫t1+ 2  4 dt  t  (1+t )  1   1  (t +1) 1+ t2  dt 1+...
Class XII Mathematics Mathematics Part-II 1 x dx ∫ −1 ∫ −1 ∫1 1 3 = x sinπ x dx− 2 xsinπ x dx ∫ −1 ∫ 1 Integrating both integrals on righthand side, we get ∫ 2| x sinπ x|dx = −1 2 − − 1 − 1 ...
Class XII Mathematics Mathematics Part-II 1  1 0 π  –1bt  π  –1au  π  –1 b –1a  π 2 = ab tan a  – ab tan b  = tan + tan  =  0  1 ab  a b  2ab Miscellaneous Exercise on Chap...
Class XII Mathematics Mathematics Part-II 1 1 cos x  ∫0 cos 4x +sin 4x ∫0 cos x 4sin x 2 π 1 dx πsinx cosx 27. 3 sinx + cosx dx 28. 29. 4 dx ∫π sin 2x ∫0 1+ x − x ∫0 9 16sin 2 x π 30. 2sin ...
Class XII Mathematics Mathematics Part-II 1 is the inverse of differentiation. Let d F(x) = f (x) . Then we write f (x) dx = F(x)+C . These integrals dx ∫ are called indefinite integrals or g...
Class XII Mathematics Mathematics Part-II 7 2026-27 PROBABILITY 433 (A) 4 (B) 1 (C) (D) 2 5 2 5 5 14. If A and B are two events such that A ⊂ B and P(B) ≠ 0, then which of the followingiscorr...