Evaluate: `int(x-5)\ sqrt(x^2+x)\ dx`

Answer

To evaluate the integral \( I = \int (x - 5) \sqrt{x^2 + x} \, dx \), we will follow a systematic approach.

### Step 1: Rewrite the Integral
We start with:
\[ I = \int (x - 5) \sqrt{x^2 + x} \, dx \]

### Step 2: Express \( x - 5 \) in terms of the derivative of \( x^2 + x \)
We can express \( x - 5 \) as a linear combination of the derivative of \( x^2 + x \). The derivative is:
\[ \frac{d}{dx}(x^2 + x) = 2x + 1 \]
We can write:
\[ x - 5 = A(2x + 1) + B \]
where \( A \) and \( B \) are constants to be determined.

### Step 3: Determine Coefficients \( A \) and \( B \)
Expanding the equation:
\[ x - 5 = (2A)x + A + B \]
Comparing coefficients, we have:
1. \( 2A = 1 \) (coefficient of \( x \))
2. \( A + B = -5 \) (constant term)

From the first equation, we find:
\[ A = \frac{1}{2} \]
Substituting \( A \) into the second equation:
\[ \frac{1}{2} + B = -5 \implies B = -5 - \frac{1}{2} = -\frac{10}{2} - \frac{1}{2} = -\frac{11}{2} \]
Thus, we have:
\[ x - 5 = \frac{1}{2}(2x + 1) - \frac{11}{2} \]

### Step 4: Substitute Back into the Integral
Now we can substitute back into the integral:
\[ I = \int \left( \frac{1}{2}(2x + 1) - \frac{11}{2} \right) \sqrt{x^2 + x} \, dx \]
This gives us:
\[ I = \frac{1}{2} \int (2x + 1) \sqrt{x^2 + x} \, dx - \frac{11}{2} \int \sqrt{x^2 + x} \, dx \]

### Step 5: Define New Integrals
Let:
\[ I_1 = \int (2x + 1) \sqrt{x^2 + x} \, dx \]
\[ I_2 = \int \sqrt{x^2 + x} \, dx \]
Thus:
\[ I = \frac{1}{2} I_1 - \frac{11}{2} I_2 \]

### Step 6: Solve \( I_1 \)
For \( I_1 \), we can use the substitution \( u = x^2 + x \), then \( du = (2x + 1) \, dx \):
\[ I_1 = \int \sqrt{u} \, du = \frac{2}{3} u^{3/2} + C = \frac{2}{3} (x^2 + x)^{3/2} + C \]

### Step 7: Solve \( I_2 \)
For \( I_2 \), we can complete the square:
\[ x^2 + x = \left(x + \frac{1}{2}\right)^2 - \frac{1}{4} \]
Then:
\[ I_2 = \int \sqrt{\left(x + \frac{1}{2}\right)^2 - \left(\frac{1}{2}\right)^2} \, dx \]
Using the formula for the integral of a square root, we find:
\[ I_2 = \frac{1}{2} \left( x + \frac{1}{2} \right) \sqrt{x^2 + x} - \frac{1}{4} \log \left| x + \frac{1}{2} + \sqrt{x^2 + x} \right| + C \]

### Step 8: Combine Results
Substituting \( I_1 \) and \( I_2 \) back into our expression for \( I \):
\[ I = \frac{1}{2} \left( \frac{2}{3} (x^2 + x)^{3/2} \right) - \frac{11}{2} \left( \frac{1}{2} \left( x + \frac{1}{2} \right) \sqrt{x^2 + x} - \frac{1}{4} \log \left| x + \frac{1}{2} + \sqrt{x^2 + x} \right| \right) \]

### Final Answer
After simplifying, we arrive at the final result for the integral \( I \).