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MATHEMATICS, INTEGRALS, Synopsis, If y=f(x) be a function and f1(x) = f(x), then, ,  f ( x)dx, , = F(x) +c, , Where ‘c’ is a constant and F(x) + c is called an indefinite integral of f(x)., Thus, Also,, (i), , d, dx, ,  f ( x)dx, , = F(x)+c , , d, F (x) =f(x), dx, ,  f ( x)dx = f(x), , (ii), , f, , 1, , ( x)dx  f ( x), , Elementary Standard integrals :, 1), 3), 5), 7), 8), 9), 10), 11), 12), 13), 14), 16), 18), 20), , x n 1,  x dx  n  1  C, n  1, , , 1, 1,  x n dx    (n  1) x n1  + C, n  1, n, , 1, 1,  x 2 dx  x  C, ax, x, a, dx, , C, , log a, , 6), , 1, , 2), ,  x dx  log x  c, , 4), , , , 1, , x, x,  e dx  e  C, , x, , dx  2 x  C, and, , ax,  e dx , , e ax, C, a, ,  cos(ax  b), C, a, sin(ax  b),  cos x dx  sin x  C similarly,  cos(ax  b) dx  a  C, 1,  tan x dx  log | sec x | C similarly,  tan(ax  b) dx  a log | sec(ax  b) | C, 1,  cot x dx  log | sin x | C similarly,  cot(ax  b) dx  a log | sin(ax  b) | C,  sec x dx  log | sec x  tan x | C  log tan  4  x 2  C,  cos ecx dx  log | cos ecx  cot x | C  log tan x 2  C, dx, dx, 1, 15) ,  log x  x 2  a 2 +C,  a 2  x 2  sin x a  C, x2  a2, dx, dx, ax, 2, 2,  1 log, C, 17),  2,  x 2  a 2  log x  x  a +C, 2a, 2, ax, a x, ,  sin x dx   cos x  C, , similarly,, ,  sin(ax  b) dx , , , ,  , , , , dx, dx, 1, 19),  1 tan 1 x  C, ,  2, 2a, a, a, 2, 2, 2, x a, a x, 2, 2, 2, 2, x, a 2 log x  x 2  a 2  C, x, , a, dx, , x, , a, , , 2, 2, 2, 2,  a  x dx, , 22), , , , I., , Method of Substitution :, If, f ( x)dx  F ( x)  C , then, , x 2  a 2 dx , , , , xa, C, xa, , 1 x a 2  x 2  a 2 sin 1 x  C, 2, 2, a, 2, 2, 2, 2, x, x  a  a log x  x  a 2  C, 2, 2, , 21), , , , log, ,  f ( x)dx  F ( x)  C, Absolute Learning Academy 1

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MATHEMATICS, ie;, , (i), , [ f ( x)]n1, C, n 1, , f 1 ( x), dx  log f ( x)  C, f ( x), , (ii), , , , (iii), , e, , (iv), , f ( x), 1,  a . f ( x)dx , , f ( x), , . f 1 ( x)dx  e f ( x )  C, , f 1 ( x), , a f ( x), C, log a, , (v), , , , (vi), ,  f ( g ( x)) . g ( x)dx,  e ( f ( x)  f ( x)dx, , (vii), II., , n 1,  [ f ( x)] f ( x)dx , , f (x ), , dx  2 f ( x)  C, 1, , x, , 1, ,  F[ g ( x)]  C,  e x f ( x)  C ., , Integration by parts :, , du, ,  (u.v)dx  u. vdx   ( vdx). dx .dx, III. Integrals of the form, ,  ax, , px  q,  bx  c, , 2, , &, , px  q, , , , \, , ax 2  bx  c, , d, (ax 2  bx  c)  B, dx, Find A&B . Split the given integral into the sum of two integrals and then find each of the, integrals., Rule : Express the numerator px+q as, , IV. Integrals of the form, ,  ax, , 2, , dx,  bx  c, , &, , A, , dx, , , , \, , ax 2  bx  c, Rule : Take out the numerical coefficient of x2 from ax 2  bx  c and complete the squares, of the terms in x., ie;, =, ax 2  bx  c = a x 2  b x  c, a, a, 2, 2, a  x  1 .coefficien t of x  cons tan t  1 coefficien t of x , 2, 2, , , , , , , , , , , , , , , , Then the given integral reduces to one of the standard forms., V., , Integration by partial fraction :, , VI. Integrals of rational functions of cosine and sine :, Any rational function of sinx and cosx can be reduced to a rational function of ‘t’ by the, , 2, Dx=, dt ,, 1 t2, , substitution t=tan x . Then, 2, Remember :, , 1, x|x| + C, 2, , 1., ,  |x| dx =, , 3., ,  sin-1x dx = xsin-1x + 1  x 2 +C,  tan-1x dx = x tan-1x  1 log(1+x2) +C, 2, 1,  cot-1x dx = x cot-1x , log(1+x2) +C, 2, , 5., 6., , 1 t2, 2t, cosx=, and sinx=, ., 2, 1 t, 1 t2, , 2.  logx dx = x log x-x +C, , , , , , 4.  cos-1x dx = x cos-1x - 1  x 2 +C, , 7., ,  sec-1x dx = x sec-1x-log x  x 2  1 +C, , 8., ,  cosec-1x dx = x cosec-1x + log x  x 2  1 +C, , , , , , Absolute Learning Academy 2

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MATHEMATICS, 16., ,  cos, , 1  1 , ,  dx ,  x, , a) x cos 1 x  sec 1 x b) x sec 1 x cosh 1 x c) x sec 1 x  cos 1 x d) x sec 1 x  cosh 1 x, 17., , tan x, ,  sin x cos xdx , 1, b) sec x  c, a) cot x  c, d) 2 tan x  c, tan x  c, 2, x,  1  sin 5 dx , x, x, x, x, 1, x, x, , ,  cos   c,  sin   c, a) 10 sin, b) 10 cos, c),  cos  sin   c, 10 , 10, 10 , 10 , 10, 10 ,  10, , 1, x, x, d),  sin  cos   c, 10  10, 10 , sin x  3 cos x,  sin x  cos x dx , a) 2 x  log(sin x  cos x)  c b) tan x  cot x  c, c) x  logsin x  cos x  c , d) logsec x  tan x   c, a), , 18., , 19., , 20., , x2, ,  x 6  2 x 3  2dx , , , , , , a) tan 1 x 3  1  c, 21., ,  sin, , 2, , b), , , , 23., , 24., , c), , , , , , 1, cot 1 x 2  1  c, 3, , , , , , d) cot 1 x 2  1  c, , x cos 6 xdx , , 1 3, 1, tan x  tan5 x  c, 3, 5, 1, 1, c)  tan3 x  tan5 x  c, 3, 5, sin 2 x, dx  tan 1 f x   c then f x , If , 4, 4, cos x  sin x, a) x 2, b) cos x, c) sin x, a) tan x , , 22., , , , 1, tan 1 x 3  1  c, 3, , cos x  x sin x, dx , x( x  cos x), 1, a)  log( x  cos x)  c, x, , 1 3, 1, tan x  tan5 x  c, 3, 5, 1, d) tan 5 x  log sec x  c, 3, b), , d) tan 2 x, , , , b) log(, , x, x  cos x, )  c c) log x( x  cos x)  c d) log(, )c, x  cos x, x, ,  [sin(log x)  cos(log x)]dx , , a) sin(log x)  cos(log x)  c b) x sin(log x)  c c) x cos(log x)  c d) sin(log x)  cos(log x)  c, 25., , x5, ,  1  x2, , dx =, , Absolute Learning Academy 5

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MATHEMATICS, , 26., , 2 3 x 32 x 5 x, c, log( 720), , a), 27., , x, , b), , 30., , 1., ,  ( x  1), , b) e x ( x 2  2 x  1)  c, ,  , ,  , ,  , , 2 3 x 32 x 5 x, c, log( 90), , d) None, d) e x ( x 2  1)  c, , d) none of these, , a) cos(b+a)log |sec(x+b) |+xsin(b-a)+C b) cos(b-a)log |sec(x-b) |+xsin(b-a)+C, c) cos(x-a)log |sec(x-b) |+xsin(x-a)+C, d) none, Home Test :, cos 5 x. cos 3x.dx =, , , , sin 5 x. sin 3x, C, 15, sin 8 x sin 2 x, c), , C, 16, 4, 2 x  11,  x 2  10 x  26 dx =, , b), , sin 8 x sin 2 x, , C, 8, 2, sin 8 x, d),  sin2x+C, 4, , b) tan-1(x2+10x+26)+log(x+5)+C, , , (x  5,   C,  ( x  10 x  26 , , x  x  1,  e  x 2  dx =, , , , e x 1  x e 1, e x  xe, x, , , , d) none of these, , 2, , a) log(x-1)+C b) xex+C, , e, , x, , e, , b) e log( e  x ), , 1  log x 2 dx , , ex, +C, x, , d) xex+C, , c), , 1, log( e x  x e ), e, , d) none, , x, , 1, (1  log x) 3, 3, dx,  x log x log(log x) , , a), , c), , dx , , a) log( e  x ), , 6., , c) e x ( x  1)  c, , b) e x x 2  c, , , , 5., , c) e x ( x 2  2 x  2)  c, ,  1  sin x ,  dx = f(x)+constant, then f(x)=,  1  cos x , a) e x cot x, b) e x cot x, c) - e x cot x, 2, 2, 2, sin( x  a),  cos( x  b) dx=, If  e x , , c) , , 4., , d), , e dx , , a) log(x2+10x+26)+tan-1(x+5)+C, , 3., , 2 3 x 32 x 5 x, c, log(180), , 2 x, , a), , 2., , c), , e dx , , a) xe x  c, 29., , 2 3 x 32 x 5 x, c, log( 360), , x 4 x3, c), ,  tan 1 x  c, 4, 3, , 2 x, , a) e x ( x 2  1)  c, 28., , x4 x2 1, b), ,  log( x 2  1)  c, 4, 2 2, , x4 x2, a), ,  tan 1 x  c, 4, 2, 4, x, x3, d), ,  tan 1 x  c, 4, 3, If  23 x 32 x 5 x dx , , b), , 1, (1  log x) 2, 2, , c) log(1  log x), , d) none, , Absolute Learning Academy 6

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MATHEMATICS, b) loglog(log x), , a) log(log x), 7., , , ,   5x, , , , , , 1, log 5 x 2  6 +C, 10, log(log x),  x log x dx , , , , b) log, , 1, log 5 x 2  6 +C, 5, , b), , 1, log(log x)2, 2, , , , 14., , 7 sin x  5 cos x, ,  3 sin x  2 cos x, , 17., , b), ,  5x , 1, tan 1  , 5,  3, , d), ,  5x , 1, tan 1  , 5,  6, , c) log(log x), , d) none, , x, 2, , d) log x  2 log sec , , , , , x, 2, , 1, x  log(sin x  cos x) c) x  log sin x    d) x  log cos x   , 2, 4, 4, , , , , , 10 x 9  10 x log e 10, x10  10 x, , If, , 1, 31, log(3sinx+2cosx)+, x+C, 13, 13, 1, 31, b), log(2sinx-3cosx)+ x+C, 12, 13, f (x) , 1 2, 1 2, c) log cos x  x, d) log cos x  x, 2, 2, b), , dx , , , , x, 10, b) 10  x, , c) 10 x  x10, , , , 1, , d) log(10 x  x10 ), ,  f ( x)dx  f ( x), then, b) f ( x)  cons tan t, , tan(log x), dx , x, a) log cos(log x), , c) f ( x)  0, , d) f ( x)  e x, , , , e, , log(cotx ), , b) log sec(log x), , x, a), , c) log sin(log x), , d) none, , dx =, , a) cosec2x+C b) cosx+c, 18., , c), , 2, c) log x sec, , 1, 31, log(3sinx-2cosx)+ x+C, 13, 13, 1, 31, c), log(2sinx-3cosx)x+C, 13, 13, If f (0)  f (0)  0 and f (x)  tan 2 x then, 1 2, 1 2, a) log sec x  x, b) log sec x  x, 2, 2, , a) f ( x)  x, 16., , 1, 1, (x log x)2+C d) (x log x)3+C, 2, 2, , dx =, , x, 10, a) 10  x, , 15., , , , x, 1  cos x, , a), , 13., , c), , sin x, ,  sin x  cos xdx , a) x  log(sin x  cos x), , 12., , b), , 1  x sin x  cos x, dx , x(1  cos x), , a) log x(1  cos x), 11., , 1, (x logx-x)2+C, 2, , x , dx =,  6 , , a) log(log x)2, 10., , b), , 2, , a), 9., , d) none, ,  xlogx(logx+1) dx is equal to, a) 2(x logx-x)2+C, , 8., , c) log(log x)2, , c) log(sinx)+C, , d) log(cosx)+C, , x 1, dx =,  8 x  11, 7, , 8, , 1, log(x 8 +8x+11)+C, 8, , b), , 1, tan-1(x 8 +8x+11)+C, 8, , c) 8 tan-1(x8+8x+11)+C, , d) none, , Absolute Learning Academy 7

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MATHEMATICS, 19., , If  (sin4x-cos4x) dx =, , 1, 2 2, , sin(4x+a)+b, then, , 5, 5, , bR, b) a= , b R, 4, 4, 2, 2, If  x 3 51/ x dx=k 51/ x , then k equal to, a) a=, , 20., , a) -2log5, 21., , cos 2 x, ,  (cos x  sin x), , 2, 2 log 5, , b) -, , 2, , , , 24., , 25., , 26., , cos x  x sin x, dx=, x( x  cos x), , c) log|cosx-sinx|+C d) none, , d) log, , x, +C, x  cos x, , cot(log x), dx , x, a) sinloglog x , b) logsin(log x), c) logcos(log x), d) none, sin x  cos x, If , dx is equal to, 9  16 sin 2 x, 1, 1, 1,  4(sin x  cos x) , a) log |sinx-cosx+, b) sin-1 , 9  16 sin 2 x1, , 5, 4, 4, 4, , , 1,  4(sin x  cos x) , c) sin-1 , d) none of these, , 5, 5, , , sin 2 x dx,  a 2 sin 2 x  b 2 cos 2 x is equal to, 2, a) log (a2 sin2x+b2cos2x), b) 2, log(a2sin2x+b2 cos2x), a  b2, 1, c) 2, log(a2sin2x+b2 cos2x), d), none of these, a  b2, , , , e, , x log a x, , e dx is equal to, b), , {1  2 tan x(tan x  sec x)}, , (ae) x, log( ae), , , , c), , ex, 1 log x, , d) none, , dx is equal to, , a) log sec x(sec x  tan x)  c, c) log sec x(sec x  tan x)  c, , b) log cos ecx (sec x  tan x)  c, d) log(sec x  tan x)  c, , sin x  cos x sin x, e, cos xdx is equal to, 1  sin 2 x, , a) e, 29., , 1, 2 log 5, , x, , ,   C c) log (x(x+cosx))+C,  x  cos x , , 12, , 28., , d) -, , b) log , , a) (ae) x, 27., , d) none of these, , dx=, , a) log |x(x+cosx)|+C, 23., , , , b R, 4, , c) 2log5, , a) –log(cosx-sinx)+C b) –log|cosx-sinx|+C, 22., , c) a=, , sin x, ,  (1  e, , x, , c, , b) e, , sin xcos x, , 1, dx, = a), +C, x, )(1  e ), 1  ex, , c, , c) e, b), , sin xcos x, , 1, +C, 1 ex, , c, c), , d) e, , 1, +C, (1  e x ) 2, , cos xsin x, , c, , d) none, , Absolute Learning Academy 8

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MATHEMATICS, 30., , x, , 1, x 6 16, , dx =, , a) sec-1 (x3)+C, , b) log (x6-16)+C, , c) 1, , 12, , sec-1(, , d) cosec-1(x3)+C, , 1, c, 11, b, 21, b, , 2, a, 12, b, 22, d, , 3, c, 13, b, 23, b, , 4, c, 14, d, 24, b, , Answer Key:, 5, 6, a, b, 15, 16, d, b, 25, 26, c, b, , 7, c, 17, c, 27, c, , x3, )+C, 4, , 8, a, 18, a, 28, a, , 9, b, 19, a, 29, b, , 10, d, 20, d, 30, c, , Absolute Learning Academy 9