JEE Exam » JEE Important Formulas » JEE Maths Formulas – Part 3

# JEE Maths Formulas – Part 3

In this article we will go through maths quick formula revision for JEE. Find the important formulas for Inverse Trigonometric Functions, Straight Line, Indefinite Integration and Application Of Derivatives.

## Inverse Trigonometric Functions formula

The formula for inverse trigonometric functions are as stated below

DescriptionFormula
Arcsine Function

Arcsine function is an inverse of sine function which is denoted by sin-1

The formula for arcsin is given as

sin-1(-x)=-sin-1(x), x∈[-1, 1]

Domain of arcsin is -1≤x≤1

Range of arcsin is -π/2≤y≤π/2

Differentiation of sin-1x is 1/√(1-x^2 )

Arccosine Function

Arccosine function is an inverse of cosine function which is denoted by cos-1

cos-1(-x)=π-cos-1(x), x∈[-1, 1]

Domain of arccos is -1≤x≤1

Range of arccos is 0≤y≤π

Differentiation of cos-1x is –1/√(1-x^2 )

Arctangent Function

Arctangent function is an inverse of tangent function which is denoted by tan-1

tan-1(-x)=-tan-1(x), x∈R

Domain of Arctangent is -∞≤x≤∞

Range of Arctangent is -π/2≤y≤π/2

Differentiation of tan-1x is 1/(1+x^2 )

Arc cotangent (Arc cot) Function

Arc cotangent function is an inverse of cotangent function which is denoted by cot-1

cot-1(-x)=π-cot-1(x), x∈R

Domain of Arc cotangent is -∞≤x≤∞

Range of Arc cotangent is 0≤y≤π

Differentiation of cot-1x is –1/(1+x^2 )

Arc secant Function

Arc secant function is an inverse of cosine function which is denoted by cot-1

sec-1(-x)=π-sec-1(x), |x|≥ 1

Domain of Arc secant is -∞≤x≤-1 or 1-∞≤x≤-∞

Range of Arc secant is 0≤y≤π, y ≠π/2

Differentiation of sec-1x is (-1)/(|x| √(x^2-1))

Arc cosecant Function

Arc cosecant function is an inverse of sine function which is denoted by cosec-1

cosec-1(-x)=-cosec-1(x), x≥ 1

Domain of Arc cosecant is -∞≤x≤-1 or 1-∞≤x≤-∞

Range of Arc cosecant is -π/2≤y≤π/2, y ≠0

Differentiation of cosec-1(x) is (-1)/(|x| √(x^2-1))

## Straight Line Formula

The formula for straight line are as stated below

DescriptionFormulas
Section Formulax=(mx_2±nx_1)/(m±n);y=(my_2±ny_1)/(m±n)
Centroid, Incentre and Excenter

Centroid G((x_1+x_2+x_3)/3 ,(y_1+y_2+y_3)/3)

In center I(ax_1+bx_2+cx_3)/(a+b+c),(ay_1+by_2+cy_3)/(a+b+c)

Excentre I1(-a_x+bx_2+cx_3)/(-a+b+c),(-ay_1+by_2+cy_3)/(-a+b+c)

Area of Triangle∆ ABC=1/2 |x_1 y_1 1 x_2 y_2 1 x_3 y_3 1 |
Slope formula

Line Joining two points (x_1 y_1 ) |&(x_2 y_2) |

m=(y_1-y_2)/(x_1-x_2 )
Condition of collinearity of three points|x_1 y_1 1 x_2 y_2 1 x_3 y_3 1 |=0
Angle between two straight linestanθ=|(m_1-m_2)/(1+m_1 m_2 )|
Bisector of the angles between two lines(ax+by+c)/√(a^2+b^2 )=±(〖(a〗^| x+b^| y+c^|))/√(〖a^’〗^2+〖b^’〗^2 )
Condition of ConcurrencyFor three lines a_1 x+a_2 y+c_1=0 , i=123 is|a_1 b_1 c_1 a_2 b_2 c_2 a_3 b_3 c_3 |=0
A pair of straight lines through originax^2+2hxy+by^2=0

If is the acute angle between the pair of straight lines, then tanθ

|(2√((h^2-ab) ))/(a+b)|
Two Lines:

ax+bx+c=0 and a’x+b’y+c’=0 Two lines

1. Parallel if a/a’=b/b’c/c’
2. Distance between two parallel lines= |(C_1-C_2)/√(a^2+b^2 )|
3. Perpendicular: if aa’+bb’=0
A point and line
1. Distance between point and line=|(ax_1+by_1+c)/√(a^2+b^2 )|
2. Reflection of a point about a line:
(x-x_1)/a=(y-y_1)/b=-2 (ax_1+by_1+c)/(a^2+b^2 )
1. Foot of the perpendicular from a point on the line is
(x-x_1)/a=(y-y_1)/b=-(ax_1+by_1+c)/(a^2+b^2 )

## Indefinite Integration formula

The formula for indefinite integration  are as stated below

If f & g are functions of x such thatg’x=f(x) then,

f(x)dx=gx+c⟺ddxgx+c=f(x)

Here, c is called the constant of integration

Standard Formula:
• ∫ (ax+b)^n dx=(ax+b)^(n+1)/a(n+1) +c,n≠-1
• ∫ dx/(ax+b)=1/a □ln ln (ax+b) +c
• ∫ e^(ax+b) dx=1/a e^(ax+b)+c
• ∫ a^(px+q) dx=1/P a^(px+q)/(□ln ln a )+c , Here a>0
• ∫ sin (ax+b)dx =-1/a □cos cos (ax+b) +c
• ∫ sin (ax+b)dx =-1/a □cos cos (ax+b) +c
• ∫ tan (ax+b)dx =1/a □ln ln □sec sec (ax+b) +c
• ∫ cot (ax+b)dx =1/a □ln ln □sin sin (ax+b) +c
• ∫(ax+b)dx =1/a tan (ax+b) +c
• ∫(ax+b)dx =-1/a cot (ax+b) +c
• ∫(ax+b)dx =-1/a cot (ax+b) +c

or∫ dx =ln tan (π/4+x/2)+c

• ∫ dx =ln (x +cot x ) +c or ∫ dx =ln tan x/2 + c
• ∫ dx =ln (cosec x+cot x ) +c
• ∫ dx =ln (cosec x+cot x ) +c
• ∫ dx/(a^2+x^2 )=-1/a x/a +c
• ∫ dx/(|x| √(x^2+a^2 ))=-1/a x/a +c
• ∫ dx/√(x^2+a^2 )=ln [x+√(x^2+a^2 )] +c
• ∫ dx/√(x^2-a^2 )=ln [x+√(x^2-a^2 )] +c
• ∫ √(a^2-x^2 ) dx=x/2 √(x^2+a^2 )+a^2/2 x/a +c
• ∫ √(x^2+a^2 ) dx=x/2 √(x^2+a^2 )+a^2/2 ln ((x+√(x^2+a^2 ))/a) +c
• ∫ √(x^2-a^2 ) dx=x/2 √(x^2-a^2 )-a^2/2 ln ((x+√(x^2-a^2 ))/a) +c
Integration by substitutionsIf we substitute fx=t, then f’xdx=dt
Integration by part∫ (f(x)g(x))dx=f(x)∫ (g(x))dx-∫ (d/dx (f(x)) ∫ (g(x))dx)dx
Integration of type∫ dx/(ax^2+bx+c),∫ dx/√(ax^2+bx+c),∫ √(ax^2+bx+c) dx

Make the substitute x+b2a=t

Integration of trigonometric functions

∫ dx/(a+bx ) or∫ dx/(a+bx )   or∫ dx/(ax +b□sin sin x □cos cos x +cx )

Here we put tan x =t

∫ dx/(a+b□sin sin x ) or∫ dx/(a+b□cos cos x ) or∫ dx/(a+b□sin sin x +c□cos cos x )

Here we put tan x/2 =t

Integration of type∫ (x^2+1)/(x^4+Kx^2+1) dx

Here k is any constant

So, we divide numerator and denominator by x^2 and put x∓1/x=t

## Application Of Derivatives formula

The formula for application of derivatives are as stated below

DescriptionFormula
Equation of tangent and normal
• Tangent at(x_1,y_1) is given by(y-y_1 )=f^’ (x_1 )(x-x_1 ), here the f'(x_1) should be real
• And normal at(x_1,y_1) is given by(y-y_1 )=-1/(f^’ (x_1 ) )(x-x_1), here the f^’ (x_1 ) should be non-zero and real.
Tangent from an external point

Given a point P(a, b) which does not lie on the curve y = f(x), then the equation of possible tangents to the curve y = f(x), passing through (a, b) can be found by solving for the point of contact Q.

f^’ (h)=(f(h)-b)/(h-a)

And equation of the tangent is

y-b=(f(h)-b)/(h-a) (x-a)
Length of tangent, normal, subtangent, subnormal
• PT=|k| √(1+1/m^2 ) is the length of the tangent
• PN=|k| √(1+m^2 )  is the length of normal
• TM=|k/m| is the length of the subtangent
• MN=|km| is the length of subnormal
Angle between the curves

Angle between two intersecting curves is defined as the acute angle between their tangents (or normal) at the point of intersection of two curves. So,

□tan tan θ =|(m_1-m_2)/(1+m_1 m_2 )|
Rolle’s Theorem:

If a function f defined on [a, b] is

• continuous on [a, b]
• derivable on (a, b) and
• f(a) = f(b),
• then there exists at least one real number c between a and b

(a < c < b) such that f’(c) = 0

Lagrange’s Mean Value Theorem (LMVT):

If a function f defined on [a, b] is

(i) Continuous on [a, b] and  (ii) derivable on (a, b)

then there exists at least one real numbers between a and b (a < c < b) such that

(f(b)-f(a))/(b-a)=f'(c)
Formulae of Mensuration
• Volume of a cuboid =lbh
• Surface area of cuboid =2(lb+bh+hl)
• Volume of cube =a3
• Surface area of cube =6a2
• Volume of a cone =1/3 πr^2 h
• Curved surface area of cone =πrl (l=slant height)
• Curved surface area of a cylinder =2πrh
• Total surface area of a cylinder =2πrh+2πr2
• Volume of a sphere =43r3
• Surface area of a sphere =4πr2
• Area of a circular sector =1/2 r^2 θ , here θ is in radian
• Volume of a prism =area of the base×(height)
• Lateral surface area of a prism =perimeter of the baseheight
• Total surface area of a prism =lateral surface area×2(area of the base)
• Volume of a pyramid =1/3area of the base×(height)
• Curved surface area of a pyramid =1/2perimeter of the base×(slant height)