If $\int_0^4 \frac{dx}{2x + 1} = \log k$, then the value of k is :

Step 1: Integrate the function

We need to evaluate the definite integral. The integral of $1/(ax+b)$ is $(1/a) \log |ax+b|$. In this case, $a=2$ and $b=1$.

Step 2: Apply the limits of integration

Now we apply the limits of integration from $0$ to $4$. This means we substitute the upper limit into the integrated function and subtract the result of substituting the lower limit.

Step 3: Evaluate the definite integral

Substitute $x=4$ and $x=0$ into the integrated expression. This gives us $\frac{1}{2} \log(9) - \frac{1}{2} \log(1)$.

Step 4: Simplify the expression

Since $\log 1 = 0$, the expression simplifies to $\frac{1}{2} \log 9$. We can also write $\frac{1}{2} \log 9$ as $\log (9^{1/2})$ which is $\log 3$.

Step 5: Find the value of k

We are given that the integral equals $\log k$. By comparing our result $\log 3$ with $\log k$, we find that $k=3$.

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