The value of the definite integral `int _(0) ^(10) ((x-5) +(x-5)^(2) +(c-5)^(3))` dx is:

Options

A. `(125)/(3)`
B. `(250)/(3)`
C. `(125)/(6)`
D. `(250)/(4)`

Answer

To solve the definite integral

\[ \int_{0}^{10} \left( (x-5) + (x-5)^{2} + (x-5)^{3} \right) dx, \]

we will follow these steps:

### Step 1: Rewrite the Integral
First, we can express the integral as:

\[ I = \int_{0}^{10} \left( (x-5) + (x-5)^{2} + (x-5)^{3} \right) dx. \]

### Step 2: Use the Property of Definite Integrals
We can use the property of definite integrals which states that:

\[ \int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a + b - x) \, dx. \]

In our case, \( a = 0 \) and \( b = 10 \). Therefore, we can write:

\[ I = \int_{0}^{10} \left( (10-x-5) + (10-x-5)^{2} + (10-x-5)^{3} \right) dx. \]

### Step 3: Simplify the Expression
Now, substituting \( 10 - x - 5 \) gives us:

\[ 10 - x - 5 = 5 - x. \]

So, we can rewrite the integral as:

\[ I = \int_{0}^{10} \left( (5-x) + (5-x)^{2} + (5-x)^{3} \right) dx. \]

### Step 4: Combine the Two Integrals
Now we have two expressions for \( I \):

1. \( I = \int_{0}^{10} \left( (x-5) + (x-5)^{2} + (x-5)^{3} \right) dx \)
2. \( I = \int_{0}^{10} \left( (5-x) + (5-x)^{2} + (5-x)^{3} \right) dx \)

Adding these two equations gives:

\[ 2I = \int_{0}^{10} \left( (x-5) + (5-x) + (x-5)^{2} + (5-x)^{2} + (x-5)^{3} + (5-x)^{3} \right) dx. \]

### Step 5: Simplify the Combined Integral
Notice that \( (x-5) + (5-x) = 0 \). Therefore, the linear terms cancel out.

Next, we simplify the squares and cubes:

\[ (x-5)^{2} + (5-x)^{2} = 2(x-5)^{2}, \]
\[ (x-5)^{3} + (5-x)^{3} = 0. \]

Thus, we have:

\[ 2I = \int_{0}^{10} 2(x-5)^{2} \, dx. \]

### Step 6: Solve for \( I \)
This simplifies to:

\[ I = \int_{0}^{10} (x-5)^{2} \, dx. \]

Now, we can perform a substitution. Let \( u = x - 5 \), then \( dx = du \). The limits change as follows:

- When \( x = 0 \), \( u = -5 \).
- When \( x = 10 \), \( u = 5 \).

Thus, we have:

\[ I = \int_{-5}^{5} u^{2} \, du. \]

### Step 7: Evaluate the Integral
Now we evaluate:

\[ \int u^{2} \, du = \frac{u^{3}}{3} \Big|_{-5}^{5} = \frac{5^{3}}{3} - \frac{(-5)^{3}}{3} = \frac{125}{3} - \left(-\frac{125}{3}\right) = \frac{125 + 125}{3} = \frac{250}{3}. \]

### Final Answer
Thus, the value of the definite integral is:

\[ \boxed{\frac{250}{3}}. \]