# Solving Mathematical Equation Involving Integration (Math Problem Sample)

Solving mathematical problems

source..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

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