integration?

Question

could somebody please help me to do these integration questions as i am very rusty:

S 7 / (5-6x) dx

S (3 - 7x) / (1 + x^2) dx

S (5x^3 - 17x^2 + 6x + 1) / (x^2 + 1)(x - 2)^2 dx

*the S is supposed to be the S shaped integration symbol if u didnt already realise*

Answer 1

I'll help you with the 3rd exercise. The first 2 are simpler, and if you understand the 3d, you automatically understood the first 2. I won't do the algebraic work, but I'll show you what you have to do.

The numerator is a polynomial of degee 3 and the numerator is a polynomial of degrre 4 > 3. We know from Algebra that the rational function (5x^3 - 17x^2 + 6x + 1) / (x^2 + 1)(x - 2)^2 can be split into 3 parcels, like that:

(5x^3 - 17x^2 + 6x + 1) / (x^2 + 1)(x - 2) = (ax + b)/(x^2 +1) + c/(x -2)^2 + d/(x-2), where a, b, c, d are constants to determine. Determining them requires a lot of algebraic work. You multiply both sides by (x^2 +1)(x -2)^2 and equal the coefficients of each side. You get a system of linear equations with 4 equations and 4 unknowns. This system has one and only one solution for a, b, c, d. Pure algebraic work.

Once a, b, c, d are determined, we can solve the integral. We solve each of the 3 parcels and sum then up. We have

S (ax + b)/(x^2 +1) dx = a/2 S (2x)/(x^2 +1) + b S dx/(x^2 +1) = a/2 ln(x^2 +1) + b arctan(x) + K1, K1 an integration constant

S c/(x-2)^2 dx = -c/(x-2) + K2 and

S d/(x-2) = d ln(x-2) + K3

Since K1 + K2 + K3 = K is a constant too, your integral is a/2 ln(x^2 +1) + b arctan(x) - c/(x-2) + d ln(x-2) + K. As you can see, the integration itself is simple, but fiding a,b, c and d, though very simple too, is kinda boring, requires a lot of work.

Well, the first 2 execises are kinda simple, I'll do them:

1)S 7 / (5-6x) dx = -7/6S 6/(6x - 5) = -7/6 ln(6x -5) + K

2) S (3 - 7x) / (1 + x^2) dx = 3 S dx/(1+x^2) - 7/2 S(2x) (x^2 + 1) dx = 3 arctan(x) - 7/2 ln(x^2 + 1) + K. Observe that 2x is the derivative of x^2 + 1.

Answer 2

for #1, assume, y=5-6x so, dy = -6dx.

S 7/y * -1/6 dy = -7/6 * S 1/y dy =-7/6 * ln(y) = -7/6 * ln(5-6x)