Math Formula

OM PRAKASH CHAPAGAIN, JHAKASH1 Matrix and Determinant 1. Determinant of a matrix: A = bc ad d c b a ÷ = A = ) ( ) ( ) ( ge dh c fg di b fh ei a h g e d c i g f d b i h f e a i h g f e d c b a ÷ + ÷ ÷ ÷ = + ÷ = 2. Equality of matrices: s r q p d c b a = , a=p, b=q, c=r, d=s 3. Scalar multiplication of matrix: kd kc kb ka d c b a k = 4. Negative of a matrix is found by multiplying it by -1. 5. Addition or subtraction: s d r c q b p a s r q p d c b a ± ± ± ± = ± , 6. Multiplication: ds cq dr cp bs aq br ap s r q p d c b a + + + + = × , 7. Transpose of a matrix (A T ): if f e d c b a A = , than, f d b e c a A T = 8. Let ij M be the matrix obtained by deleting the th i row and th j column of matrix A. The determinant ij M is called a minor of the matrix A. The scalar ij j i ij M A + ÷ = ) 1 ( is called the cofactor of the element ij a of the matrix A. 9. Vector product(dot product): cf be ad f e d c b a f e d c b a + + = · = 10. Ad joint of a matrix ( ) ( ) T ij A A adj = . OM PRAKASH CHAPAGAIN, JHAKASH 2 11. Inverse of a matrix: A Adj A A . 1 1 = ÷ , 1 1 1 = = ÷ ÷ A A AA , If B A = ÷1 than A B = ÷1 12. A square matrix is said to be symmetric if the ij j i ) , ( element of the matrix is equal to the th i j ) , ( element, for all values of and. ji ij A A = . A square matrix is said to be skew symmetric if the th j i ) , ( element is equal to the negative of the th i j ) , ( element, for all values of and. ji ij A A ÷ = . 13. Conjugate: The Matrix obtained from any given Matrix A by replacing its elements by the corresponding conjugate complex numbers is called the conjugate of A and denoted by A. If iy x i iy x A ÷ + = 2 3 than, iy x i iy x A + ÷ ÷ = 2 3 14. Properties of matrices: i) ( ) ( ) C B A C B A ± ± = ± ± ii) C A B B A = ± = ± (A new matrix) iii) T T T B A AB = ) ( iv) ( ) KB KA B A K ± = ± v) 0 = = IA AI vi) A A = ± 0 vii) 0 ) ( = ÷ + A A viii) ( ) AC AB C B A ± = ± ix) 0 0 = A x) If n is a positive integer then n A n A A A A = · · · ......... , 2 A A A = · xi) ) ( ) )( ( AB adj adjB adjA = xii) 1 1 1 1 1 ) ( ÷ ÷ ÷ ÷ ÷ · = · = A B B A AB xiii) T T T B A B A + = + ) ( xiv) I A A A adj A adj A = = ) . ( ) . ( 15 Properties of Determinant: i) The value of the determinant is unaltered if its rows and columns are interchanged. ) det( ) det( T A A = . OM PRAKASH CHAPAGAIN, JHAKASH 3 ii) If two adjacent rows or columns are interchanged, the sign of the determinant changes, but its numerical value remains unaltered. iii) If two rows or columns of a determinant are identical (all elements zero), the value of the determinant is zero. iv) If all the elements of one row or one column be multiplied by a non-zero constant k, then the value of the determinant is multiplied by k. v) If each element of a row or column is expressed as the sum of two numbers, then the determinant can be expressed as a sum of two determinants of the same order. vi) A determinant is unaltered in value, by adding to all the elements of any column or any row the same multiple of the corresponding elements of any number of other columns or rows. vii) In any determinant, if the elements of any row or column are multiplied by the cofactors of the corresponding elements of any other row or column, the sum of the products would be equal to zero. viii) A common factor of all the elements of a row of A can be taken outside the determinant. The same applies to columns. ix) If all the elements of A below (or above) the diagonal are zero then the determinant is equal to the product of the diagonal elements. In particular, the determinant of a diagonal matrix is equal to the product of the diagonal elements. x) The determinant of a product is the product of the determinants. Symbolically, B A AB det det ) det( · = OM PRAKASH CHAPAGAIN, JHAKASH 4 16. Cramer’s Rule for solving simultaneous equation: If, 3 3 3 3 2 2 2 2 1 1 1 1 d z c y b x a d z c y b x a d z c y b x a = + + = + + = + + then let 3 3 3 2 2 2 1 1 1 c b a c b a c b a = A , 3 3 3 2 2 2 1 1 1 2 c b d c b d c b d = A , 3 3 3 2 2 2 1 1 1 2 c d a c d a c d a = A and 3 3 3 2 2 2 1 1 1 3 d b a d b a d b a = A . If 0 = A , A A = 1 x , A A = 2 y , A A = 3 z 17. Solve equation without expending (determinant method) by making unit matrix. For example: x y x d c b a d c b a d c b a = 3 3 3 3 2 2 2 2 1 1 1 1 : : : make like this z y x r q p = : 1 0 0 : 0 1 0 : 0 0 1 and the answer is x=p, y=q, z=r. It applies for 2×2 matrices also. Quadratic Equation 1. Structure: c bx ax x f + + = 2 ) ( if 0 = a is a standard form of a quadratic equation. ) )( ( ) ( 2 1 r x r x a x f ÷ ÷ = is factored form, where 1 r and 2 r are the roots of the quadratic equation. k h x a x f + ÷ = 2 ) ( ) ( is a vertex form, where ) , ( k h are the ) , ( y x coordinates of the vertex. 2. Formula to find the value of x, a ac b b x 2 4 2 ÷ ± ÷ = 0 = a i) If 0 4 2 > ÷ ac b , then roots are real and distinct. ii) If 0 4 2 = ÷ ac b , then roots are real and equal. iii) If 0 4 2 < ÷ ac b , then roots are imaginary and distinct.therein OM PRAKASH CHAPAGAIN, JHAKASH 5 3. Let α and β be the roots of the equation. Then sum of roots a b ÷ = + | o and the product of roots a c = o| . 4. i) Quadratic equations with real coefficients the complex roots always occur in conjugate pairs. i.e., | o i + and | o i ÷ . ii) Quadratic equations with rational coefficients the irrational roots always occur in conjugate pairs. i.e., | o + and | o ÷ . 5. Formation of equation from the given roots α and β. The equation is: 0 ) ( 2 = + + ÷ o| | o x x Trigonometric Functions 1. u u sin ) sin( ÷ = ÷ 2. u u cos ) cos( = ÷ 3. u u tan ) tan( ÷ = ÷ 4. u u csc ) csc( ÷ = ÷ 5. u u sec ) sec( = ÷ 6. u u cot ) cot( ÷ = ÷ 7. u u u cos sin tan = 8. u u u sin cos cot = 9. u u csc 1 sin = 10. u u sec 1 cos = 11. u u cot 1 tan = 12. 1 cos sin 2 2 = + u u 13. u u 2 2 csc cot 1 = + 14. u u 2 2 sec tan 1 = + 15. u u u cos sin 2 2 sin = 16. 1 cos 2 sin 2 1 sin cos 2 cos 2 2 2 2 ÷ = ÷ = ÷ = u u u u u 17. u u u 3 sin 4 sin 3 3 sin ÷ = 18. u u u u u 2 2 sin 2 1 2 sin tan 1 tan 2 2 tan ÷ = ÷ = 19. B A B A B A sin cos cos sin ) sin( ± = ± 20. B A B A B A sin sin cos cos ) cos(  = ± 21. B A B A B A tan tan 1 tan tan ) tan(  ± = ± OM PRAKASH CHAPAGAIN, JHAKASH 6 22. u u cos 3 cos 4 3 cos 3 ÷ = 23. u u t cos ) 2 sin( = ÷ 24. u u u u 2 3 tan 3 1 tan tan 3 3 tan ÷ ÷ = 25. u u t sin ) 2 cos( = ÷ 26. u u t cot ) 2 tan( = ÷ 27. u u t tan ) 2 cot( = ÷ 28. u u t csc ) 2 sec( = ÷ 29. u u t sec ) 2 csc( = ÷ 30. u u t cos ) 2 sin( = + 31. u u t sin ) 2 cos( ÷ = + 32. u u t cot ) 2 tan( ÷ = + 33. u u t tan ) 2 cot( ÷ = + 34. u u t csc ) 2 sec( ÷ = + 35. u u t sec ) 2 csc( = + 36. x x sin ) sin( = ÷ t 37. x x cos ) cos( ÷ = ÷ t 38. x x tan ) tan( ÷ = ÷ t 39. x x cot ) cot( ÷ = ÷ t 40. x x sec ) sec( ÷ = ÷ t 41. ( ) x x csc csc = ÷ t 42. x x sin ) sin( ÷ = + t 43. x x cos ) cos( ÷ = + t 44. x x tan ) tan( = + t 45. x x cot ) cot( = + t 46. x x sec ) sec( ÷ = + t 47. ( ) x x csc csc ÷ = + t 48. x x sin ) 2 sin( = + t 49. x x cos ) 2 cos( = + t 50. x x tan ) 2 tan( = + t 51. x x cot ) 2 cot( = + t 52. x x csc ) 2 csc( = + t 53. x x sec ) 2 sec( = + t 54. x x sin ) 2 sin( ÷ = ÷ t 55. x x cos ) 2 cos( = ÷ t 56. x x tan ) 2 tan( ÷ = ÷ t 57. x x cot ) 2 cot( ÷ = ÷ t 58. x x sec ) 2 sec( = ÷ t 59. x x csc ) 2 csc( ÷ = ÷ t 60. x x cos ) 2 3 sin( ÷ = ÷ t 61. x x sec ) 2 3 csc( ÷ = ÷ t 62. x x sin ) 2 3 cos( ÷ = ÷ t 63. x x csc ) 2 3 sec( ÷ = ÷ t 64. x x cot ) 2 3 tan( = ÷ t 65. x x tan ) 2 3 cot( = ÷ t 66. x x cos ) 2 3 sin( ÷ = + t 67. x x sin ) 2 3 cos( = + t 68. x x sec ) 2 3 csc( ÷ = + t 69. x x csc 2 3 sec = | . | \ | + t 70. x x cot 2 3 tan ÷ = | . | \ | + t 71. x x tan 2 3 cot ÷ = | . | \ | + t 72. 2 cos 2 sin 2 sin x x x = 73. 2 cos 1 2 sin x x ÷ = 74. 2 cos 1 2 cos x x + = 75. 2 tan 1 2 tan 2 tan 2 x x x ÷ = 76. 2 sin 2 cos 2 sin 2 1 1 2 cos 2 cos 2 2 2 2 x x x x x ÷ = ÷ = ÷ = 77. x x x x x x x cos 1 sin sin cos 1 cos 1 cos 1 2 tan + = ÷ = + ÷ = 78. 2 cos 2 sin 2 sin sin D C D C D C ÷ + = + 79. 2 sin 2 cos 2 sin sin D C D C D C ÷ + = ÷ 80. 2 cos 2 cos 2 cos cos D C D C D C ÷ + = + OM PRAKASH CHAPAGAIN, JHAKASH 7 81. 2 sin 2 sin 2 cos cos C D D C D C ÷ + = ÷ where C=A+B and D=A-B and 2 D C A + = and 2 B C B ÷ = 82. 2 sec 2 tan 1 | 2 csc 2 cot 1 | 1 2 cos 2 sin 2 2 2 2 2 2 x x x x x x = + = + = + Properties of triangle 1. Sine law: R C c B b A a 2 sin sin sin = = = This implies that a <= b <= c if and only if A <= B <= C. 2. Cosine law: A i) A bc c b a cos 2 2 2 2 ÷ + = ii) B ac c a b cos 2 2 2 2 ÷ + = c b iii) C ab b a c cos 2 2 2 2 ÷ + = 3. Tangent law: B a C i) 2 tan 2 tan B A B A b a b a ÷ + = ÷ + ii) 2 tan 2 tan A C A C a c a c ÷ + = ÷ + ii) 2 tan 2 tan C B C B c b c b ÷ + = ÷ + 4. Newton’s formulae: i) 2 sin 2 cos C B A c b a ÷ = + ii) 2 sin 2 cos A C B a c b ÷ = + iii) 2 sin 2 cos B A C b a c ÷ = + 5. Mollweide’s Formulae: i) 2 cos 2 sin C B A c b a ÷ = ÷ ii) 2 cos 2 sin A C B a c b ÷ = ÷ iii) 2 cos 2 sin B A C b a c ÷ = ÷ 6. bc A A = 2 sin , ac B A = 2 sin , ab C A = 2 sin , ) )( )( ( c s b s a s s ÷ ÷ ÷ = A where 2 c b a s + + = OM PRAKASH CHAPAGAIN, JHAKASH 8 7. 2 sin 2 sin 2 sin B ca A bc C ab = = = A , C B A c B A C b A C B a sin 2 sin sin sin 2 sin sin sin 2 sin sin 2 2 2 = = = A 8. C c B b A a abc R sin 2 sin 2 sin 2 4 = = = A = 9. bc c s b s A ) )( ( 2 sin ÷ ÷ = 10. bc a s s A ) ( 2 cos ÷ = 11. ) ( ) )( ( 2 tan c s s c s b s A ÷ ÷ ÷ = 12. ac c s a s B ) )( ( 2 sin ÷ ÷ = 13. ab b s a s C ) )( ( 2 sin ÷ ÷ = 14. ac b s s B ) ( 2 cos ÷ = 15. ) ( ) )( ( 2 tan b s s c s a s B ÷ ÷ ÷ = 16. ab c s s C ) ( 2 cos ÷ = 17. ) ( ) )( ( 2 tan c s s b s a s C ÷ ÷ ÷ = 18. c b a s + + = 2 , A = rs 19. 2 tan ) ( 2 sin 2 cos 2 sin 2 sin ) )( )( ( C c s s C ab C B A c s c s b s a s s r ÷ = = = ÷ ÷ ÷ = A = 20. Other relations: i) B c C b a cos cos + = ii) A c C a b cos cos + = iii) A b B a c cos cos + = iv) Semi-perimeter 2 ) ( c b a s + + = s v) r = radius of inscribed circle, vi) R= radius of circumscribed circle vii) A=Area of a triangle. A OA=OB=OC=R Q P O OP=OQ=OR=r C B R Big circle drawn outside the triangle is called a circumscribed circle and small circle inside the triangle is called an inscribed circle. 21. R r C B A + = + + 1 cos cos cos OM PRAKASH CHAPAGAIN, JHAKASH 9 Complex Numbers 1. 1 1 2 ÷ = ¬ ÷ = i i . A complex number is any number in the form of bi a + , where ‘a’ and ‘b’ are the real numbers. 2. The conjugate of a complex number bi a + is written as bi a + and is equal to bi a ÷ . (Similarly bi a ÷ is equal to bi a + .) 3. For a complex number iy x z + = , we define absolute value z as being the distance from ‘z’ to ‘0’ in the complex plane ‘z’. Thus 2 2 2 y x z + = , 2 2 x x = & 2 2 y y = . Therefore 2 2 y x z + = . 4. Addition: i d b c a di c bi a ) ( ) ( ) ( ) ( + + + = + + + 5. Subtraction: i d b c a di c bi a ) ( ) ( ) ( ) ( ÷ + ÷ = + ÷ + 6. Division: 2 2 2 2 2 2 2 2 ) ( ) ( ) 1 ( ) 1 ( ) ( d c i ad bc bd ac d c bd i bc ad ac i d c bdi bci adi ac di c bi a + ÷ + + = ÷ ÷ ÷ ÷ ÷ ÷ = ÷ ÷ + ÷ = + + 7. Multiplication: i bc ad bd ac bd i bc ad ac i bd bci adi ac di c bi a ) ( ) ( ) ( ) ( ) ( ) ( 2 + + ÷ = ÷ + + = + + + = + · + 8. Multiplication of conjugates: 2 2 2 2 2 ) )( ( b a i b a bi a bi a + = ÷ = ÷ + 9. Powers of i : 1. 1 = n i if ‘n’ is multiple of 4. 2. i i n = if ‘n’ is one more than a multiple of 4. 3. 1 ÷ = n i if ‘n’ is two more than a multiple of 4. 4. i i n ÷ = if ‘n’ is three more than a multiple of 4. 10. De Moivre’s theorem: ( ) | | ) sin (cos sin cos u u u u n i n r i r n n + = + and if ) sin (cos u u i r z + = then ) 2 sin 2 (cos 1 1 n k i n k r z n n u t u t + + + = where ) 1 .........( 3 , 2 , 1 , 0 ÷ = n k 11. Properties of magnitude: OM PRAKASH CHAPAGAIN, JHAKASH 10 2 1 2 1 z z z z = · , 2 1 2 1 z z z z = and 2 1 2 1 z z z z + > + 12. Euler’s formula: x i x e ix sin cos + = and ix x i x In = + ) sin (cos 13. Cube roots and properties of unity: ω 3 =1, 2 3 1 i w ± ÷ = i) Each of the complex roots of unity is square of the other. 2 2 2 ) ( 1 w w = = . In general, 3 1 2 = + + n n w w (if n is a multiple of 3) and 0 1 2 = + + n n w w (if n is an integer but not a multiple of 3) ii) Sum of cube roots of unity is zero. 0 1 2 = + + w w where w , 1 and 2 w are the cube roots of unity. 14. Complex argument: It is written as Arg(z) where z is a positive number. Different values of Arg are given below: i) Arg(1) = 0 ii) Arg(1+i) = iii) Arg(i) = iv) Arg(-i) = v) Arg(-1) = vi) ) ( tan ) ( 1 y x iy x Arg ÷ = + Inverse Trigonometric Functions 1. u u 1 sin csc 1 1 ÷ ÷ = 2. u u 1 cos sec 1 1 ÷ ÷ = 3. u u 1 tan cot 1 1 ÷ ÷ = 4. 2 cos sin 1 1 t u u = + ÷ ÷ 5. 2 cot tan 1 1 t u u = + ÷ ÷ 6. 2 sec csc 1 1 t u u = + ÷ ÷ 7. u u 1 1 sin ) ( sin ÷ ÷ ÷ = ÷ 8. u t u 1 1 cos ) ( cos ÷ ÷ ÷ = ÷ 9. u u 1 1 tan ) ( tan ÷ ÷ ÷ = ÷ 10. u u 1 1 csc ) ( csc ÷ ÷ ÷ = ÷ 11. u t u 1 1 sec ) ( sec ÷ ÷ ÷ = ÷ 12. u u 1 1 cot ) ( cot ÷ ÷ ÷ = ÷ 13. u u u u u u u u 1 csc 1 1 sec 1 cot 1 tan 1 cos sin 1 2 1 2 1 2 1 2 1 1 ÷ ÷ ÷ ÷ ÷ ÷ = ÷ = ÷ = ÷ = ÷ = OM PRAKASH CHAPAGAIN, JHAKASH 11 14. 2 1 1 2 1 2 1 2 1 1 1 1 csc 1 sec 1 cot 1 tan 1 sin cos u u u u u u u u ÷ = = ÷ = ÷ = ÷ = ÷ ÷ ÷ ÷ ÷ ÷ 15. u u u u u u u u 1 cot 1 sec 1 csc 1 1 cos 1 sin tan 1 2 1 2 1 2 1 2 1 1 ÷ ÷ ÷ ÷ ÷ ÷ = + = + = + = + = 16. | | 2 2 1 1 1 1 1 sin sin sin x y y x y x ÷ + ÷ = + ÷ ÷ ÷ 20. x x x 1 2 1 tan 2 1 2 sin ÷ ÷ = + 17. | | 2 2 1 1 1 1 1 sin sin sin x y y x y x ÷ ÷ ÷ = ÷ ÷ ÷ ÷ 21. x x x 1 2 2 1 tan 2 1 1 cos ÷ ÷ = + ÷ 18. | | 2 2 1 1 1 1 1 cos cos cos y x xy y x ÷ ÷ ÷ = + ÷ ÷ ÷ 22. x x x 1 2 1 tan 2 1 2 tan ÷ ÷ = ÷ 19. | | 2 2 1 1 1 1 1 cos cos cos y x xy y x ÷ ÷ + = ÷ ÷ ÷ ÷ 23. xy y x y x  1 tan tan tan 1 1 1 ± = ± ÷ ÷ ÷ , for positive and for negative 24. If x y 1 sin ÷ = then Limits and Derivatives Formulas In the following, u and v are functions of x, and n, e, a, and k are constants. 1. h x f h x f h x f ) ( ) ( 0 lim ) ( ' ÷ + ÷ = The Definition of the Derivative. 2. 0 ) ( = k dx d The derivative of a constant is zero. 3. ( ) { } | | dx du k x u k dx d ) The derivative of a constant times a function. 4. ( ) dx du nu u dx d n n 1 ÷ = The Power Rule (Variable raised to a constant). OM PRAKASH CHAPAGAIN, JHAKASH 12 5. ( ) dx dv dx du v u dx d + = + The Sum Rule. 9. dx du du dv dx dv · = The Chain Rule. 6. ( ) dx dv dx du v u dx d ÷ = ÷ The Difference Rule. 7. dx du v dx dv u vu uv v u dx d + = + = ' ' ) . ( ÷The Product Rule. ÷ dx du vw dx dv uw dx dw uv dx dy + + = 8. 2 2 ' ' ) ( v dx dv u dx du v v uv vu v u dx d ÷ = ÷ = The Quotient Rule. 10. | | { } ) ( ' ) ( ' )} ( { x g x g f x g f dx d · = Another Form of the Chain Rule. 11. dx du u u dx d cos ) (sin = The Derivative of the Sine. 12. dx du u u dx d sin ) (cos ÷ = The Derivative of the Cosine. 13. dx du u u dx d 2 sec ) (tan = The Derivative of the Tangent. 14. dx du u ec u dx d 2 cos ) (cot ÷ = The Derivative of the Cotangent. 15. dx du u u u dx d tan sec ) (sec = The Derivative of the Secant. 16. dx du u ecu ecu dx d cot cos ) (cos · ÷ = The Derivative of the Cosecant. 17. dx du u u dx d 2 1 1 1 ) (sin ÷ = ÷ The Derivative of the Inverse Sine. 18. dx du u u dx d 2 1 1 1 ) (cos ÷ ÷ = ÷ The Derivative of the Inverse Cosine. 19. dx du u u dx d 2 1 1 1 ) (tan + = ÷ The Derivative of the Inverse Tangent. OM PRAKASH CHAPAGAIN, JHAKASH 13 20. dx du u u dx d 2 1 1 1 ) (cot + ÷ = ÷ The Derivative of the Inverse Cotangent. 21. dx du u u u dx d 1 1 ) (sec 2 1 ÷ = ÷ The Derivative of the Inverse Secant. 22. dx du u u u ec dx d 1 1 ) (cos 2 1 ÷ ÷ = ÷ The Derivative of the Inverse Cosecant. 23. dx du u Inu dx d 1 ) ( = The Derivative of the Natural Log. 24. dx du Ina u u dx d a · = 1 ) (log The Derivative of the log to base a. 25. dx du e e dx d u u = ) ( The Derivative of e raised to a variable. 26. dx du Ina a a dx d u u = ) ( The Derivative of a constant raised to a variable. 27. 0 0 lim = A A ÷ A = x y x dx dy 28. The reciprocal of dy dx is the derivative dx dy of the direct function. 29. 0 1 cos 0 lim = A ÷ A ÷ A x x x 30. 0 sin 0 lim = A A ÷ A x x x 31. Limit Calculation: a. | | ) ( lim ) ( lim x f a x c x cf a x ÷ = ÷ b. | | ) ( lim ) ( lim ) ( ) ( lim x g a x x f a x x g x f a x ÷ ± ÷ = ± ÷ c. | | ) ( lim ) ( lim ) ( ) ( lim x g a x x f a x x g x f a x ÷ · ÷ = ÷ d. | | n n x f a x x f a x ÷ = ÷ ) ( lim ) ( lim OM PRAKASH CHAPAGAIN, JHAKASH 14 e. ) ( lim ) ( lim ) ( ) ( lim x g a x x f a x x g x f a x ÷ ÷ = ÷ if 0 ) ( lim = ÷ x g a x f. · = · ÷ x e x lim and 0 lim = ÷· ÷ x e x g. · = ±· ÷ r x x lim for even r h. · = · ÷ ) ( lim x In x and ÷· = ÷ + ) ( 0 lim x In x i. If r > 0 then 0 lim = · ÷ r x c x j. · = · ÷ r x x lim and ÷· = ÷· ÷ r x x lim for odd r. j. If r > 0 and r x is real for x < 0 then 0 lim = ÷· ÷ r x c x Integration Formulae 1. C x dx + = í 2. C ax dx a dx a + = = í í 3. C n x dx x n n + + = + í 1 1 4. C w v u dw dv du dw dv du + + + = + + = + + í í í í ) ( 5. C x In dx x + = í 1 6. C e dx e x x + = í 7. 0 , > + = í a C a In a dx a x x 8. C x xdx + ÷ = í cos sin 9. C x xdx + = í sin cos 10. C x xdx + = í tan sec 2 11. C x xdx x + = í sec tan sec 12. C x xdx x + ÷ = í csc cot csc 13. C x xdx + ÷ = í cot csc 2 14. 1 1 1 1 1 1 + ÷ + = + = í + + + n a n b n x xdx n n b a n b a 15. dx vdx u dx d vdx u vdx u í í í í ÷ = ) ( . OM PRAKASH CHAPAGAIN, JHAKASH 15 16. Integration by substitution method. A. 2 2 x a ÷ , use u sin a x = B. 2 2 x a + , use u tan a x = C. 2 2 a x ÷ , use u sec a x = 17. For dx x x n m í cos sin , Case 1, ‘n’ is odd. Suppose n is odd. Hence, . So that we split , whose power is . And let find the derivatives of and solve it. For example: xdx x í 3 2 cos sin xdx x x I cos cos sin 2 2 í = Let, xdx du x dx du x dx d dx du cos cos sin = = = Now, substitute the value of xdx cos in the equation and simplify it. í í ÷ = ÷ = du u u I xdx x x I ) 1 ( cos ) sin 1 ( sin 2 2 2 2 5 3 5 3 4 2 u u I du u du u I ÷ = ÷ = í í Again substitute the value of u which is the final answer. 5 sin 3 sin 5 3 x x I ÷ = Case 2, suppose ‘m’ is odd, m=2k+1, so that we split sinx whose power is m. And let u=cosx, find the derivatives of cosx and solve it. For x u tan , set ) 1 ) ( )(sec ( tan ) ( tan ) ( tan ) ( tan 2 2 2 2 ÷ = = ÷ ÷ x x x x x n n n OM PRAKASH CHAPAGAIN, JHAKASH 16 For ) ( cot x n , set ) 1 ) ( )(csc ( cot ) ( cot ) ( cot ) ( cot 2 2 2 2 ÷ = = ÷ ÷ x x x x x n n n For ) ( sec x n , set ) 1 ) ( )(tan ( sec ) ( sec ) ( sec ) ( sec 2 2 2 2 + = = ÷ ÷ x x x x x n n n For ) ( csc x n , set ) 1 ) ( )(cot ( csc ) ( csc ) ( csc ) ( csc 2 2 2 2 + = = ÷ ÷ x x x x x n n n Partial Fractions 1. 3 5 3 ) 3 )( 5 3 ( 1 ÷ + ÷ = ÷ ÷ ÷ x B x A x x x 2. 2 3 3 ) 2 3 )( 3 ( 1 2 2 + + + + + = + + + ÷ x x C Bx x A x x x x 2 2 ) 1 ( 1 3 ) 1 )( 3 ( 1 . 3 + + + + ÷ = + ÷ ÷ x C x B x A x x x