# Integral of (2x+5) (x^2-3) / x - Moshe

• nmoshe
In summary, the integral of (2x+5)(x^2-3)/x is equal to (x^3-3x^2+5x-15)ln(x) + C. To solve this integral, you can use integration by parts or polynomial long division to separate the rational function into two separate integrals. The constant term "C" represents the constant of integration and is necessary to include in the solution to account for all possible solutions to the integral. No, substitution cannot be used to solve this integral as there is no single function or variable that can be substituted to simplify the expression. Moshe is not a variable or function in this integral. It is likely a typo or placeholder name used for the problem
nmoshe
hi guys,

i need help with following eqation.

(2x+5) (x^2-3)
--------------- << entire equation is divide by x.
x

i need integral value of this equation.

moshe

You also need help with notation. This is not an integral "equation" because you have no equation- no equal sign. You appear to be asking for the integral
$$\int \frac{(2x+5)(x^2-3)}{x}dx$$

Looks pretty straight forward to me: (2x+5)(x2-3)= 2x3+ 5x2- 6x- 15 so
$$\frac{(2x+5)(x^2-3)}{x}= 2x^2+ 5x- 6-\frac{15}{x}$$.

Can you integrate that?

Hi Moshe,

To solve this integral, we can use the partial fraction decomposition method. First, let's rewrite the equation as:

(2x+5) (x^2-3) / x = (2x^3 + 5x^2 - 6x - 15) / x

Next, we can factor out the x from the numerator to get:

(2x^3 + 5x^2 - 6x - 15) / x = x (2x^2 + 5x - 6 - 15/x)

Now, we can perform partial fraction decomposition on the term 15/x. This means we can write it as:

15/x = A/x + B

where A and B are constants that we need to solve for. Multiplying both sides by x, we get:

15 = A + Bx

To solve for A and B, we can plug in some values for x. Let's choose x = 0 and x = 1. This gives us the equations:

15 = A + 0, so A = 15

and

15 = A + B, so B = 0

Now, we can rewrite the original equation as:

(2x+5) (x^2-3) / x = x (2x^2 + 5x - 6 - 15/x) = x (2x^2 + 5x - 6 - 15/x) = x (2x^2 + 5x - 6 - 15/x) = x (2x^2 + 5x - 6 - 15/x) = x (2x^2 + 5x - 6 - 15/x) = x (2x^2 + 5x - 6 - 15/x) = 2x^3 + 5x^2 - 6x - 15

Now, we can integrate each term separately. The integral of 2x^3 is (1/2)x^4 + C, the integral of 5x^2 is (5/3)x^3 + C, the integral of -6x is -3x^2 + C, and the integral of -15 is -15x + C. Combining these, we get the final answer:

∫ (2x+5) (x^2-

## 1. What is the integral of (2x+5)(x^2-3)/x?

The integral of (2x+5)(x^2-3)/x is equal to (x^3-3x^2+5x-15)ln(x) + C.

## 2. How do you solve the integral of (2x+5)(x^2-3)/x?

To solve this integral, you can use integration by parts or polynomial long division to separate the rational function into two separate integrals.

## 3. What is the purpose of the constant term "C" in the solution?

The constant term "C" represents the constant of integration and is necessary to include in the solution to account for all possible solutions to the integral.

## 4. Can the integral of (2x+5)(x^2-3)/x be solved using substitution?

No, substitution cannot be used to solve this integral as there is no single function or variable that can be substituted to simplify the expression.

## 5. Who was Moshe and what is their significance in this integral?

Moshe is not a variable or function in this integral. It is likely a typo or placeholder name used for the problem.

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