`int_(0)^(1) ( x^(4) + 1)/( x^(2) + 1) dx` is equal to

Options

A. `(1)/( 6) ( 3- 4pi)`
B. `(1)/( 6) ( 3pi + 4)`
C. `( 1)/( 6) ( 3 + 4pi )`
D. `(1)/( 6) ( 3pi - 4)`

Answer

To solve the integral \( I = \int_{0}^{1} \frac{x^4 + 1}{x^2 + 1} \, dx \), we can simplify the integrand and break it down into manageable parts. Here’s the step-by-step solution:

### Step 1: Rewrite the integrand
We start with the integral:
\[ I = \int_{0}^{1} \frac{x^4 + 1}{x^2 + 1} \, dx \]
We can rewrite the numerator \( x^4 + 1 \) as \( (x^4 + 2x^2 + 1) - 2x^2 \):
\[ I = \int_{0}^{1} \frac{(x^2 + 1)^2 - 2x^2}{x^2 + 1} \, dx \]
This simplifies to:
\[ I = \int_{0}^{1} (x^2 + 1) \, dx - 2 \int_{0}^{1} \frac{x^2}{x^2 + 1} \, dx \]

### Step 2: Evaluate the first integral
Now we evaluate the first integral:
\[ \int_{0}^{1} (x^2 + 1) \, dx = \int_{0}^{1} x^2 \, dx + \int_{0}^{1} 1 \, dx \]
Calculating these separately:
\[ \int_{0}^{1} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{1} = \frac{1}{3} \]
\[ \int_{0}^{1} 1 \, dx = [x]_{0}^{1} = 1 \]
Thus,
\[ \int_{0}^{1} (x^2 + 1) \, dx = \frac{1}{3} + 1 = \frac{4}{3} \]

### Step 3: Evaluate the second integral
Next, we need to evaluate the second integral:
\[ \int_{0}^{1} \frac{x^2}{x^2 + 1} \, dx \]
We can simplify this integral:
\[ \frac{x^2}{x^2 + 1} = 1 - \frac{1}{x^2 + 1} \]
Thus,
\[ \int_{0}^{1} \frac{x^2}{x^2 + 1} \, dx = \int_{0}^{1} 1 \, dx - \int_{0}^{1} \frac{1}{x^2 + 1} \, dx \]
Calculating these:
\[ \int_{0}^{1} 1 \, dx = 1 \]
And,
\[ \int_{0}^{1} \frac{1}{x^2 + 1} \, dx = \left[ \tan^{-1}(x) \right]_{0}^{1} = \tan^{-1}(1) - \tan^{-1}(0) = \frac{\pi}{4} - 0 = \frac{\pi}{4} \]
Thus,
\[ \int_{0}^{1} \frac{x^2}{x^2 + 1} \, dx = 1 - \frac{\pi}{4} \]

### Step 4: Combine the results
Now we can substitute back into our expression for \( I \):
\[ I = \frac{4}{3} - 2\left(1 - \frac{\pi}{4}\right) \]
This simplifies to:
\[ I = \frac{4}{3} - 2 + \frac{\pi}{2} = \frac{4}{3} - \frac{6}{3} + \frac{\pi}{2} = -\frac{2}{3} + \frac{\pi}{2} \]

### Final Result
Thus, the final result for the integral is:
\[ I = \frac{\pi}{2} - \frac{2}{3} \]