The integral of 7+x-2x²÷2-x
To find the integral of the expression, we will first simplify it:
7 + x - 2x² ÷ 2 - x = 7 + x - x² - x = 7 - x²
Now, we can integrate this expression. The integral of -x² is -1/3x³, and the integral of 7 is 7x.
Therefore, the integral of 7 + x - 2x² ÷ 2 - x is:
∫ (7 - x²) dx = ∫ 7 dx - ∫ x² dx
= 7x - 1/3x³ + C
where C is the constant of integration.
To find the integral of the given expression, we can use the basic rules of integration. However, before proceeding, I noticed a potential ambiguity in the expression you provided. It is important to properly denote the division and multiplication operations to avoid confusion. Let's assume that the expression is:
∫ (7 + x - 2x²) / (2 - x) dx
To integrate this expression, we can follow these steps:
Step 1: Simplify the expression inside the integral (if possible).
- In this case, the expression is already simplified.
Step 2: Factorize the denominator (2 - x) if possible.
- Since the denominator is already in the factored form, we can skip this step.
Step 3: Decompose the expression into partial fractions.
- To proceed with this step, we need to find the roots of the denominator (2 - x) by setting it equal to zero.
2 - x = 0
x = 2
- Since the denominator has a linear factor (2 - x), the decomposition will be of the form:
(7 + x - 2x²) / (2 - x) = A / (2 - x)
- To find the value of A, we can substitute x = 2 into the original expression and equate it to A / (2 - x):
7 + 2 - 2(2)² = A / (2 - 2)
7 - 4 = A / 0 (division by zero is undefined)
- Since division by zero is undefined, the expression (7 + x - 2x²) / (2 - x) cannot be decomposed into partial fractions.
Therefore, it seems that there is an issue with the given expression. Please review the expression and make sure that it is written correctly, or provide additional information if necessary.
To find the integral of the expression 7+x-2x² ÷ (2-x), we can use the method of partial fractions. Here are the steps to solve it step-by-step:
Step 1: Rewrite the expression using long division:
Divide 2x² by (2-x): -(2x²)/(2-x) = -1
Rewrite the expression: 7+x - 1/(2-x)
Step 2: Express the denominator as a product of linear factors:
(2-x) = -1(x-2)
Step 3: Express the expression using partial fractions:
Rewrite the expression: 7+x - 1/(-1(x-2))
Separate into two fractions: 7+x + 1/(x-2)
Step 4: Determine the value of A and B in the partial fractions:
(7+x + 1/(x-2)) = A/(2-x) + B/(x-2)
Multiply both sides by (2-x)(x-2):
7+x + 1/(x-2) = A(x-2) + B(2-x)
Substitute in x = 2:
7 + 2 + 1/(2-2) = A(2-2) + B(2-2)
9 = 0
We can see that there is no value of A and B that can satisfy this equation.
Therefore, the integral of 7+x-2x² ÷ (2-x) does not have a simple closed-form solution using elementary functions.