Question

Please answer this question​

Answer 1

`\bold{\huge{\underline{ Solution }}}`

- I have used different colours for indicating different process

- I used brown colour to indicate steps

- I used green colour for explaining solution

- Pink colour for known properties of integration

- Purple colour for the middle steps that are optional.

[Note :- For solving such questions it is mandatory that you should know all the functions and it's derivatives ]

Required Answer for the given question

`\sf{=}{\sf{( x^(2))/(2)}}{\sf x - 3 }`

Answer 2

`\displaystyle \large + C`

Explanation:

To find an integration of this function, first, you must know these integration methods and differences of them.

• Partial Fraction Method - A method that separates the denominator into two brackets and solve the equation for two variables.
• Long Division Method - A method that uses long division to rewrite the fraction to make it easier to integrate.

These methods above are common when it comes to integrating fractional functions, except there are differences when to use these methods.

• Partial Fraction technique has to be used when the degree of numerator is lower than the degree of denominator. Basically, proper fraction.

Examples (of these integrands that require partial fraction technique)

`\displaystyle \large{\int (3)/(x^2-5x+6) \ dx}\\\displaystyle \large{\int (x+2)/(x^2-4) \ dx}`

• Long Division technique has to be used when the degree of numerator is greater than the degree of denominator.

Examples (of these integrands that require long division)

`\displaystyle \large{\int (x^2+5x+6)/(x+2) \ dx}\\\displaystyle \large{\int (x^4+3)/(x) \ dx }`

Therefore, from the given integral, the integrand requires long division method. (See attachment for long division.)

After long division, we should get:

`\displaystyle \large{\int x+7+(37x-81)/(x^2-7x+12) \ dx}`

Recall important properties of integral such as:

`\displaystyle \large{\int [f(x) \pm g(x)] \ dx = \int f(x) \ dx \pm \int g(x) \ dx}`

Hence:

`\displaystyle \large{\int x \ dx + \int 7 \ dx + \int (37x-81)/(x^2-7x+12) \ dx}`

Recall important integration formula for polynomial and constant:

`\displaystyle \large{\int x^n \ dx = (x^(n+1))/(n+1) + C}\\\displaystyle \large{\int x \ dx = (x^2)/(2) + C}\\\displaystyle \large{\int k \ dx = kx + C \ \ \tt{(k \ \ is \ \ a \ \ constant.)}`

Therefore:

`\displaystyle \large{(x^2)/(2)+7x+\boxed{\int (37x-81)/(x^2-7x+12) \ dx}}`

From the boxed integral above, we cannot evaluate it by default. From what I said, if the degree of numerator is less than the degree of denominator, we’ll use partial fraction technique.

Therefore, factor the denominator:

`\displaystyle \large{(37x-81)/((x-4)(x-3))}`

Set to A/x-4 and B/x-3

`\displaystyle \large{(37x-81)/((x-4)(x-3)) = (A)/(x-4) + (B)/(x-3)}`

Multiply both sides by (x-4)(x-3).

`\displaystyle \large{(37x-81)/((x-4)(x-3)) \cdot (x-4)(x-3)= (A)/(x-4) \cdot (x-4)(x-3) + (B)/(x-3) \cdot (x-4)(x-3)}\\\displaystyle \large{37x-81= A(x-3) + B(x-4)}\\\displaystyle \large{37x-81= Ax-3A+Bx-4B}\\\displaystyle \large{37x-81= (A+B)x-(3A+4B)}`

Then compare the coefficients:

`\displaystyle \large{\left \{ {{A+B=37} \atop {3A+4B=81}} \right}`

Solve the simultaneous equation for A and B.

`\displaystyle \large{\left \{ {{A=37-B} \atop {3A+4B=81}} \right}\\\displaystyle \large{\left \{ {{A=37-B} \atop {3(37-B)+4B=81}} \right}\\\displaystyle \large{\left \{ {{A=37-B} \atop {111-3B+4B=81}} \right}\\\displaystyle \large{\left \{ {{A=37-B} \atop {111+B=81}} \right}\\\displaystyle \large{\left \{ {{A=37-B} \atop {B=81-111}} \right}\\\displaystyle \large{\left \{ {{A=37-B} \atop {B=-30}} \right}\\\displaystyle \large{\left \{ {{A=67} \atop {B=-30}} \right}\\`

Therefore, A = 67 and B = -30. Substitute the values in:

`\displaystyle \large{\int (67)/(x-4)-(30)/(x-3) \ dx}\\\displaystyle \large{\int (67)/(x-4) \ dx -\int (30)/(x-3) \ dx}`

Recall the integration formula for above:

`\displaystyle \large\int (k)/(x-a) \ dx = \int k \cdot (1)/(x-a) \ dx = k\ln`

Therefore:

`\displaystyle \largex-4`

Back from this:

`\displaystyle \large{(x^2)/(2)+7x+\boxed{\int (37x-81)/(x^2-7x+12) \ dx}}`

Substitute in:

`\displaystyle \large-30\ln`

Therefore the solution is:

`\displaystyle \large \boxedx-3`