Solving Mathematical Equation Involving Integration (Math Problem Sample)

Calculus
Question 1:
To evaluate ʃ x sin 3x dx, we use the method of integration by parts given by the formula;
ʃ u dv = uv-ʃ v du
Let u = x, dv = sin 3x; then, v= -cos 3x and du = dx. Now substituting in the formula we get;
ʃ x sin 3x dx = x-1/3cos 3x-ʃ-1/3cos 3x dx = -1/3cos 3x+1/9sin 3x+C, where C is a constant.
We rearrange and obtain;
ʃ x sin 3x dx = 1/9sin 3x-1/3xcos 3x+C
Question 2:
Given that, we need to look for values of α for which A is an identity matrix. We know that A is an identity matrix iff;
Cos α=1 and –sin α = 0
Now we also know that;
Cos 0 = 1 and sin 0 = 0.
Therefore, the values of α for which matrix A is identity are α = 0
Question 3:
Given that y=mx+1 is a tangent to the curve y2 = 4x, to find the value of m, we substitute y=mx+1 in y2 = 4x and get;
(mx+1)2 = 4x →m2x2+2mx+1=4x→ m2x2+2mx-4x = -1
m2(x+2)-4 = -1/x
m2 = -1/x {1/(x-2)} = -1/(x2-2x)
m = √ {-1/(x2-2x)}
Question 4:
To solve the differential equation (x2 − y2) dx + 2 xydy = 0, we open the bracket and obtain;
x2 dx+y2 dx+2xy dy = 0
x2 dx +d(xy2) = 0 (by the fact that the differential equation of xy2 =d(xy2) = y2 dx + 2xy dy)
We then integrate x2 dx + d (xy2) = 0 on both sides and get
x3/3+xy2 = C or x3+3xy2 = C ,where C is a constant.
Question 5:
Given the system of linear equations;
x − y +2z =7 3x +4y −5z =−5 2x − y + 3z = 12
To solve the system using matrix, we rewrite the system as follows:
22288503302000014382753302000034290033020000