The value of `int_0^1(8log(1+x))/(1+x^2)dx` is a.`pilog2` b. `\ pi/8log2` c.`\ pi/2log2` d. `log2`

Options

A. `log 2`
B. `pi log 2`
C. `(pi)/8 log 2`
D. `(pi)/2 log 2`

Explanation

To solve the integral \( I = \int_0^1 \frac{8 \log(1+x)}{1+x^2} \, dx \), we will use a substitution and properties of integrals. Here’s a step-by-step solution:

### Step 1: Substitution
Let \( x = \tan(\theta) \). Then, we have:
\[ dx = \sec^2(\theta) \, d\theta \]
The limits change as follows:
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