Class name Subject Book title Chapter number Content Created at
Class XII Mathematics Mathematics Part-II 1 it is often necessary to satisfy an additional condition which then determines a specific value of C giving unique antiderivativeofthegivenfunction...
Class XII Mathematics Mathematics Part-II 1 (D) 2x + x2+C d 3 3 22. If dx f (x) = 4x − x4 such that f(2) = 0. Then f(x) is 4 1 129 3 1 129 (A) x + 3 − (B) x + 4 + x 8 x 8 4 1 129 3 1 129 (C) ...
Class XII Mathematics Mathematics Part-II 1 make the substitution mx = t so that mdx = dt. 1 1 1 Therefore, ∫ sin mx dx = ∫sin t dt= – cos t + C = – cos mx + C m m m 2 2 (ii) Derivative of x ...
Class XII Mathematics Mathematics Part-II 1 These will be used later without reference. (i) ∫tan x dx = log sec x +C We have sin x ∫tan x dx = ∫ dx cos x Put cos x = t so that sin x dx = – dt...
Class XII Mathematics Mathematics Part-II 1 1 (i) ∫sin x cos x dx (ii) ∫ sin (x + a)dx (iii)∫ 1+ tan x dx Solution (i) We have sin x cos x dx = sin x cos x (sin x) dx ∫ ∫ (1– cos x) cos x (si...
Class XII Mathematics Mathematics Part-II 1 + C 1 + 1 cos x –sin x dx = 2 2 2 ∫ cos x +sin x ... (1) Reprint 2026-27 240 MATHEMATICS cos x – sin x Now, consider I =∫ dx cos x +sin x Put cos x...
Class XII Mathematics Mathematics Part-II 1 3sin x 1 cos x 24. 25. 2 2 26. 6cos x + 4sin x cos x (1– tan x) x cos x 27. sin 2x cos 2x 28. 1+ sin x 29. cot x log sin x sin x sin x 1 30. 1+ cos...
Class XII Mathematics Mathematics Part-II 1 2  5  1 1 = – 10 cos5x + 2 cos x + C (iii) From the identity sin 3x = 3 sin x – 4 sin x, we find that 3sin x – sin 3x sin x = Therefore, ∫sin x d...
Class XII Mathematics Mathematics Part-II 1 sin2 x− cos2 x 23. ∫ 2 2 dx is equal to sin xcos x (A) tan x + cot x + C (B) tan x + cosec x + C (C) – tan x + cot x + C (D) tan x + sec x + C x 24...
Class XII Mathematics Mathematics Part-II 1 = 1 (a+ x)+(a− x)  1  1 1  a – x 2 2a  (a + x) (a − x) =  +    2a − x a+ x  Reprint 2026-27 INTEGRALS 245 dx 1  dx + dx  Therefore, ∫...
Class XII Mathematics Mathematics Part-II 1 2 = log x + x + a − log|a|+C 1 = log x + x +a +C 2 , where C = C – log |a| Applying these standard formulae, we now obtain some more formulae which...
Class XII Mathematics Mathematics Part-II 1 dx (ii) dx ∫ x −16 ∫ 2x− x 2 Solution (i) We have dx = dx = 1 log x –4 +C [by 7.4 (1)] ∫ x2 −16 ∫ x – 4 2 8 x+ 4 (ii) Put x – 1 = t. Then dx = dt. ...
Class XII Mathematics Mathematics Part-II 1 6 6 1 6x −4 = log +C 1 17 6x +30 1 3x−2 1 1 = 17 log x 5 +C 1+17 log3 1 log 3x −2 +C C + 1 log 1 = 17 x +5 , where C = 1 17 3 Reprint 2026-27 INTEG...
Class XII Mathematics Mathematics Part-II 1 1 dx = ∫ 2 2 2 x+ 3 +   2   Put x + =t , so that dx = dt, we get 1 dt 1 tan 2 tC I2= 2∫ 1 2 = 1 2 [by7.4(3)] t +  2×   2 –1  3 –1 = t...
Class XII Mathematics Mathematics Part-II 1 dt. Therefore, I = dt = sin–1 t+ C2 [by 7.4 (5)] 2 ∫ 3 −t 2 3 – 1x +2 = sin +C 2 ... (3) Substituting (2) and (3) in (1), we obtain x+3 2 –1x + C 1...
Class XII Mathematics Mathematics Part-II 1 –19x−8  1 sin–19x −8 +C (C) 3sin  8  +C (D) 2  9    7.5 IntegrationbyPartialFractions Recall that a rational function is defined as the r...
Class XII Mathematics Mathematics Part-II 1 and C are real numbers to be determined suitably. Reprint 2026-27 254 MATHEMATICS dx Example 11 Find ∫(x+1) (x +2) Solution The integrand is a prop...
Class XII Mathematics Mathematics Part-II 1 ofd constant terms on both sides, we getA+ B = 5 and 3A+ 2B = 5. Solving these equations, we get A= – 5 and B = 10 x +1 1− 5 + 10 Thus, x –5x +6 = ...
Class XII Mathematics Mathematics Part-II 1 2026-27 256 MATHEMATICS x 2 Example 14 Find ∫ 2 2 dx (x +1) (x + 4) x 2 Solution Consider 2 2 and put x = y. (x +1) (x + 4) x2 y Then 2 2 = (x +1) ...
Class XII Mathematics Mathematics Part-II 1 – 2 = A + B [by Table 7.2 (2)] (y – 2)2 y −2 (y −2)2 Therefore, 3y – 2 = A (y – 2) + B Comparing the coefficients of y and constant term, we getA= ...
Class XII Mathematics Mathematics Part-II 1 2 dx + ∫ 2 dx (x +1) (x +2) 5 x +2 5 x + 1 5 x + 1 3 1 2 1 –1 = log x+ 2 + log x +1 + tan x+C 5 5 5 EXERCISE7.5 Integrate the rational functions in...
Class XII Mathematics Mathematics Part-II 1 +1) +C 2 2 1 2 1 2 (C) −log x + log (x +1) + C (D) 2 log x +log (x +1) +C 7.6 IntegrationbyParts Inthissection,wedescribeonemoremethodofintegration...
Class XII Mathematics Mathematics Part-II 1 integral∫x cos x dx is reduced to the comparatively more complicated integral having more power of x. Therefore, the proper choice of the first fun...
Class XII Mathematics Mathematics Part-II 1 sin x dx 1− x 2 x Solution Let first function be sin– x and second function be . 1− x 2 First we find the integral of the second function, i.e., x ...
Class XII Mathematics Mathematics Part-II 1 (x)]dx = ∫ e f (x) dx + ∫ f (x) dx I + e f (x) dx, where I = e f (x) dx = 1 ∫ 1 ∫ ... (1) Taking f(x) and e as the first function and second functi...