index of problems  with some classification  --------------index
integration

integral of sqrt(tanx)     -----> (integral of sqrt(tanx))

integral of coshx cosx       --------> (integral of cosh x cosx)

integral of (1+x) / x (x-2)^0.5       --------> (integral of (1+x) / x (x-2)^0.5)

integral of ln(sin x) from 0 to pi/2  ----------> (integral of ln(sin x) from 0 to pi/2)

integral of (e^(ax))cosbx using integration by parts  ----------> (integration by parts)

 .∫ (x²) sin x  dx using integration by parts  ----------> (integration by parts)

integral of { sqrt{1+x^2}  } / {x^2} using integration by parts  ----------> (integration by parts)

∫ arc(tan4x) dx  ----------> (integration by parts)


∫ (sin3x)^4 dx  ----------> (integration by trignometric manipulation)

.∫ (x^4)(ln x)² dx ----------> (integration by parts)

∫ sec³x dx  ----------> (integration)

integrate 1/ (x^7 - x) -----------integration by adjusting

evaluate integral of (x ² +3x) with limits 1 to 3 using summation (limit of a sum)
answer: (62/3) explanation of integration by limit of sum question

evaluate integral of  [ x*exp(x)]  / (x+1) ² dx
answer: exp(x) / (x+1) + C explanation on integration by parts of exp(x)[f(x) + f '(x)] form
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area

area bounded between the curves y=x^2+2 and y=3x   --------> (area between two curves)

area under one arch of the cycloid x = a(t-sint) , y =a(1-cost) ----------problem on  area by integration

area between y²= x and x²=y     -------------area between y²= x and x²=y

find the area of the region  {(x,y) / x ² +y ² <= 1 <= x+y}
answer: (pi/4) - (1/2) more explanation on this area question

area of the cardioid r = a(1 - cosθ ) in polar form   ----------- area in polar form

other  problems on applications of integration like area ,volume etc
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differential equations


solve cot x (dy/dx) + y = csc x, with y(pi/3) = 0   --------> (linear differential equation)

solve x(dy/dx) + (1+x)y =2  (linear equation) --------> (linear equation)

solve y' -y =0 using power series method ------------>(power series method)

solve y"-y = xe^(3x)      -------------------higher order differential equation

solve cos ² x (dy/dx) + y  = tanx  answer : y = tanx - 1 + C exp(-tanx)   explanation
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mathematical induction

prove that (x-y) is a factor of (x^n -y^n ) ----------mathematical induction

proof of n^2 < 2^n for every integer n>=5     -----> (mathematical induction problem)

proof of 1^3 +2^3 + 3^3 +...+ n^3 = [ ( n (n+1) ) /2 ] ^2  -----> (mathematical induction problem)

1.2.3 + 2.3.4 + ... +n(n+1) (n+2) = n(n+1)(n+2)(n+3)/4 --------mathematical induction

4 +8 +12 + ... +4n = 2n ² + 2n  -----------------------------mathematical induction
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partial fractions
put into partial fraction 1 / {(x^3) + 1 }     -----> (partial fractions problem)

put 1/[ (x)(x-2)(2x-1)] into partial fractions ----------partial fraction

partial fraction (x^2+7x+3)/( (x^2)(x+3)) -----------partial fraction

partial fraction decomposition of (5x^3 + 8x^2 + 36x + 36) / (x^4 + 9x^2) ------partial fraction

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trigonometry
prove that (cotA+cosecA-1)/(cotA-cosecA+1)=cosecA+cotA    -----> (problem on trigonometry )

Prove that (tanxsinx) / (tanx+sinx) = (tanx-sinx) / (tanx sinx)    -----> (problem on trigonometry )

find exact value of  tan[22½ °]   or tan (pi / 8 )      -----> (problem on trigonometry )

find exact value of tan15° without using calculator --------value of tan15° without using calculator

conditional identity : If A+B+C=π,
prove that: sinA+sinB+sinC = cot(A/2).cot(B/2) [sinA+sinB-sinC] -----> (problem on conditional identity (trigonometry) )


graphs of trigonometric functions-----> (graphs of trigonometric functions )

prove that
 { tan((π/4)+x) - tan((π/4)-x)} / { tan((π/4)+x) + tan((π/4)-x) } = sin2x ---------trigonometry problem

prove that  (cos²x - sin²x) / (cos²x + sinxcosx) = 1 - tanx -------------trigonometry problem

show that
tan(x)/[1-cot(x)] + cot(x)/[1-tan(x)]= 1+sec(x)csc(x) , x≠ n π /4     ---------trigonometry

cos20° * cos40° * cos80°  without using calculator  trigonometry problem without using calculator

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implicit differentiation
if sqrt(x) +sqrt(y) = 8, find dy/dx      ----------->( implicit differentiation )

differentiation from first principles
differentiate sqrt(x) from first principles     -----> (differentiation by first principles)

if y = x sinx , find dy/dx from first principles  answer: dy/dx = xcosx +sinx explanation of first principle
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example on chain rule
differentiate ln(x + sqrt(x^2-1)) w.r.t.x     -----> ( derivative of ln(x + sqrt(x^2-1)))
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example on uvw rule
find dy/dt if y = t exp(-t) [Acost +Bsint] -----------uvw rule
rateof change
lamppost-shadow problem ------>( lamppost-shadow problem)
maxima minima
show that the semivertical angle of a right circular cone
of maximum volume and given slant height is tan ֿ¹(√2 )
explanation of maxima / minima problem

tangents and normals
find the equation of the tangent at t = pi/3 on the curve x=2cost , y = 3sint
answer: 3x + 2ysqrt(3) =12  explanation of finding equation of tangent

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example on maclaurin's series
maclaurin's series for sec(x)     -----> (maclaurin's series for sec(x))
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invese laplace transforms
inverse laplace transform of  s / [(s+3)2 +1]    -----> (inverse laplace transorm problem)

inverse laplace transform of  1/ (s^2 + 1) ( s^2 - 2s + 7)    -----> (inverse laplace transorm problem)

find L[(cosat -cosbt)/t]----------------------- laplace transform problem

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limits
limit of (x^2-49)/(x^2+5x-14) as x-->( -7)    -----> (limit)

lim [e^(-ax)-e^(-bx)]/x   as x->0 -----> (limit)

limit (1-cosx) / x² as x-->0 ------- limt using trigonometry
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sequence and series

Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P^2 R^n= S^n   -----> (g.p question)


Express  the recurring decimal 0.484848... as a fraction using the concept of geometric series  (infinite g.p question)
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matrices

Solve by matrix inversion
x – y + 2z = 13
2x + 2y – z = -6
-x + 3y + z = -7  -----> (solution by matrix inversion)

solve 3x+4y = 17 , -4x+3y =44 using matrix inversion
answer: x = -5; y =8  explanation on solving system of equations using matrix inversion


problem on matrices

answer: (x= -1) explanation of matrix problem

problem on determinant

explanation of determinant problem



solve x+y = 2 ; 2y-z = 0 ; -x-y+z = -1 using cramer's rule (using determinants)
answer : x = 3/2 , y =1/2 , z=1 explanation on solution of equations using cramer's rule





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remainder

When a polynomial p(x) is divided by (x+1), the remainder is 4. And when it is divided by (x-2), it's remainder is 3. What is the remainder  when p(x) is divided by x²-x-2 -----> (a problem on remainders)

find the quotient and remainder when x³ - 2x ² + 5x - 6  is divided by (x-3) using synthetic division-----> (a problem on long division)

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hyperbolic functions
show that sinhֿ¹x = ln (x + sqrt(x^2 + 1 )) -------------- sinhֿ¹x

show that tanhֿ¹x = x + (x³/3) +  ...    for lxl<1---------power series for inverse hyperbolic tanh


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problems on mean and variance
find the mgf  of the binomial distribution and hence deduce the mean and variance      mgf of the binomial distribution

find the mean of the poisson distribution ------mean of the poisson distribution

memory less property of geometric distribution ----------memory less property of geometric distribution

find the mean and variance of the discrete uniform distribution f(x) = 1/k , x = 1,2,...k  mean and variance of discrete uniform

show that var(aX+bY)=a²var(X)+b²var(Y) , if X and Y are independent ---------variance
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complex variables
if  u = y / (x² +y² ) , find the analytic funtion f(x) = u  + iv and hence find v ------explanation of finding analytic function
fourier series
half range fourier cosine series of f(x) = x²  in (0,l)    -------------explanation

graphical solution of a system of simultaneous linear equations in two unknowns
solve graphically  3x-7y+10=0 ,  2x-y+3=0    graphical solution

LCM of polynomials
find the  lcm of ( x³ + x² + x + 1 )  and ( x³ -  x² + x - 1 )  lcm of polynomials




(more problems)


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