Question to List Formulas for Volume of Geometrical Figures

Topic : Volume Formulas

Question : List the Formula for all kind of Geometrical figures.

Solution :

Problem to Find Unknown Values of Angles of Triangles

Topic : Angles of Triangle

Question to Indicate Order of Place Value with with an Example

Topic : Place Value

Question : Indicate Order of Place Value in a Number.

Solution :

A Simple Integration Problem

Topic : Integration

Problem : Integrate x^3 ln x



Solution :

Solved Differential Equation

Topic : Differential Equation

Problem : If y = 4 Sin 3x prove y" + 9y = 0

Solution :
y = 4 Sin 3x
differentiate with respect to x
y' = 4 Cos 3x . 3
y' = 12 Cos 3x
Again differentiate with respect to x
y" = 12 (- Sin 3x . 3)
y" = - 36 Sin 3x

Now consider y" + 9y = 0
Substitute the value of y and y"
-36 Sin 3x + 9(4 Sin 3x)
-36 Sin 3x + 36 Sin 3x = 0
Hence proved

Topic : Integration

Question : Solve by Integration using Trignometric Values
∫Cos³ 2x Sin² 2x dx

Solution :
∫Cos³ 2x Sin² 2x dx
∫Cos² 2x Sin² 2x(Cos 2x) dx

Now Sin 2x = 2 Sin x Cos x
Sin 4x = 2 Sin 2x Cos 2x
Sin² 4x = 4 Sin² 2x Cos² 2x
Multiplying and dividing by 4

¼ ∫4 Sin² 2x Cos² 2x(Cos 2x) dx
¼ ∫ Sin² 4x Cos 2x dx
Now Cos 2x = 1 – 2 Sin² x
2 Sin² x = Cos 2x – 1
Sin² x = (Cos 2x – 1) / 2
Sin² 4x = (Cos 8x -1) / 2

So,
¼ ∫ [(Cos 8x -1) / 2] Cos 2x dx
1/8 ∫ (Cos 8x Cos 2x - Cos 2x) dx
1/8 [ ∫ (Cos 8x Cos 2x ) dx - ∫ Cos 2x dx ]

Now 2 Cos A Cos B = Cos (A+B) + Cos (A-B)
1/8 [ ∫ Cos 10x + Cos 6x dx - ∫ Cos 2x dx ]
1/8 {[ (Sin 10x)/10 + (Sin 6x)/6] – (Sin 2x)/2}
1/8 [(3 Sin 10x + 5 Sin 6x – 15 Sin 2x) / 30]
3 Sin 10x + 5 Sin 6x – 15 Sin 2x) / 240

Problem on Multistep Equation

Topic : Multistep Equation

Problem : Solve 7x + 2 = -54

Solution :
7x + 2 = -54
7x = -54-2
7x = -56
dividing both sides by 7
7x/7 = -56/7
x=-8
So the solution is x = -8