Class name Subject Book title Chapter number Content Created at
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 1 where P(x) and Q(x) are polynomials in x and Q(x) ≠ 0. If degree of the polynomial P(x) is greater than the degree of the polynomial Q(x), then we ...
Class XII Mathematics Mathematics Part-II 1 −a2 2 +C (v) ∫ 2 2 =sin +C ∫ x2 − a2 a − x a dx 2 2 (vi) ∫ 2 2 = log| x+ x + a |+ C x +a ® Integration by parts For given functions1f and 2 , we ha...
Class XII Mathematics Mathematics Part-II 1 q = A d (ax +bx +c)+ B= A (2ax +b)+B , whereAand B are dx determined by comparing coefficients on both sides. b ® We have defined ∫ a f (x) dxas th...
Class XII Mathematics Mathematics Part-II 2 292 MATHEMATICS Chapter 8 APPLICATIONOFINTEGRALS ❖ One should study Mathematics because it is only through Mathematics that nature can be conceived...
Class XII Mathematics Mathematics Part-II 2 = f (x) as the result of adding up the elementary areas of thin strips across the region PQRSP. Symbolically, we express b b b A = ∫a dA = ∫a ydx =...
Class XII Mathematics Mathematics Part-II 3 or or or (Why?) Replacing v by y , we get x or 1  y2 y  2 −12 y + x or log  2 + +1 x = 3tan  +C 1 2  x x   3x  2 2 −12y + x  or log (...
Class XII Mathematics Mathematics Part-II 2 dx 2 2 2 2 2 Fig 8.5 Since x + y = a gives y = ± a − x AstheregionAOBAliesinthefirstquadrant,yistakenaspositive.Integrating,weget the whole area en...
Class XII Mathematics Mathematics Part-II 2 shown in the Fig 8.8, the area of the ellipse is b ab 2 2 = 4 ∫0xdy = 4 ∫ b − y dy (Why?) b 0 =  2   4a b×0+ b sin 1 0  Fig 8.8 = b  2 2 ...
Class XII Mathematics Mathematics Part-II 2 x = 0 and x = 2π. SolutionFromtheFig8.10,therequired area = area of the region OABO + area of the region BCDB + area of the region DEFD. Fig 8.10 T...
Class XII Mathematics Mathematics Part-II 2 ∫ a ∫ a ® The area of the region bounded by the curve x = φ (y), y-axis and the lines y = c, y = d is given by the formula:Area= d xdy = dφ(y)dy . ...
Class XII Mathematics Mathematics Part-II 2 Lastly, it is worth mentioning the following quotationbyLieSophie’s: “Itmaybesaidthattheconceptionsofdifferentialquotientandintegralwhich intheiror...
Class XII Mathematics Mathematics Part-II 3 300 MATHEMATICS Chapter 9 DIFFERENTIAL EQUATIONS ❖ He who seeks for methods without having a definite problem in mind seeks for the most part in va...
Class XII Mathematics Mathematics Part-II 3 but at this stage we shall confine ourselves to the study of ordinary differential equations only. Now onward, we will use the term ‘differential e...
Class XII Mathematics Mathematics Part-II 3 each are of degree one, equation (10) is of degree two while the degree of differentialequation(11)isnotdefined. Note Order and degree (if defined)...
Class XII Mathematics Mathematics Part-II 3 equation 2 3 2  d y  + dy  +sin dy  +1= 0 is  dx2  dx  dx  (A) 3 (B) 2 (C) 1 (D) notdefined 12. The order of the differential equation ...
Class XII Mathematics Mathematics Part-II 3 of the given differential equation. Whereas function φ contains no arbitrary constants but only the particular values of the parameters a and b and...
Class XII Mathematics Mathematics Part-II 3 sin x) = 0 = R.H.S. Therefore, the given function is a solution of the given differential equation. EXERCISE9.2 IneachoftheExercises1to10verifythat...
Class XII Mathematics Mathematics Part-II 3 (y) . g(x) ... (2) dx If h(y) ≠ 0, separating the variables, (2) can be rewritten as dy = g(x) dx ... (3) h(y) Integrating both sides of (3), we ge...
Class XII Mathematics Mathematics Part-II 3 given differential equation can be written as dy 2 = – 4x dx ... (1) y Integrating both sides of equation (1), we get dy = − 4 x dx ∫ y 2 ∫ 1 2 or ...
Class XII Mathematics Mathematics Part-II 3 The notation due to Leibnitz is extremely flexible and useful in many calculation and formal transformations,where,wecandealwithsymbolsdyanddx exac...
Class XII Mathematics Mathematics Part-II 3 2 5. (e + e ) dy – (e – e ) dx = 0 6. dx =(1+ x )(1+ y ) dy 7. y log y dx – x dy = 0 8. x5 = − y5 dx dy = sin x x x 2 9. dx 10. e tan y dx + (1 – e...
Class XII Mathematics Mathematics Part-II 3 20. In a bank, principal increases continuously at the rate of r% per year. Find the value of r if Rs 100 double itself in 10 years (loge2 = 0.6931...