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 {(2x+1) / (4+x.²)} --------->(integral of {(2x+1) / (4+x.²)})

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)/

∫ (4x - 7) / (x² + x +1) dx -----------completion of squares

definite integral ∫ x³ e^(-x)dx with limits 0 to infinity --------->(integral)

evaluate integral of { e^(1 /x^2)} / {x^3} -------------(integration by substitution)

evaluate the integral of [e^(-x) +1] ^2 ------------- integration after expansion using identity

  integral of e^{x^(1/3)}--------------  e^{x^(1/3)}

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area

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

area of the standard ellipse (x² /a²) + (y²/b²) =1 using integration ----->(area of ellipse)

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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 (D^4 - D) y = 0    ------------------ solution of homogeneous diff. equation


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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)

Prove that [1/(1x2) ]+[1/(2x3)]+...+ [1/( n(n+1) )] = n/n+1  ----mathematical induction

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discrete math (logic...)
show that (p v q) → r ≡ (p → r) ^ (q→ r)   --------->(logical reasoning)

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partial fractions
put into partial fraction 1 / {(x^3) + 1 }     -----> (partial fractions problem)

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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 )

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 )

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Inverse trigonometric functions
show that  arctanx +arctany = arctan {(x+y)/(1-xy)} --------------inverse trigonometric functions


diff. arctan √ [(1-x)/(1+x)] w.r.t.x  ---------------diff. by trig. substitution

derivative of (ax+b)/(cx+d) --------------quotient rule
implicit differentiation
if sqrt(x) +sqrt(y) = 8, find dy/dx      ----------->( implicit differentiation )

if √(xy) = x - 2y, find dy/dx     ---------------->( implicit differentiation )

  if  xy + y^2 =1 , find dy/dx  ---------------- implicit differentiation

if  y = x^(lnx) , find dy/dx ----------------logarthmic differentiation

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

find the derivative of f(x) =1/x at x=3 using the lim {f(x)-f(a)} / {x-a} as x---->a  -----------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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rate of change
lamppost-shadow problem ------>( lamppost-shadow problem)
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example on maclaurin's series
maclaurin's series for sec(x)     -----> (maclaurin's series for sec(x))

power series for arctan(x)     -----> (power series for tan ֿ¹(x))

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partial differentiation
if u =ln{sqrt (x ² +y ² )} , find ∂u/∂x ,∂u/∂y,∂²u/∂x² ,∂²u/∂x∂y ---------partial differentiation
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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)

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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)
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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)

express 0.6363... as a fraction using the concept of infinite geometric series ------------------infinite gp
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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)

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straight lines

find the cartesian and  parametric equations of the line joining (7, 2, 2) and (5, 6, -1).  -------straight line

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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quadratic equation
solve x^2 -18x +45 =0 by completing the square --------------completion of square

find the vertex of y = x^2 + 4x + 3
find the vertex of y = x^2 + 4x + 3 ---------------vertex of parabola









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