If ∫042x+1dx=logk, then the value of k is :
Get the complete, step-by-step math solution for: "If _0^4 (dx)/(2x + 1) = log k, then the value of k is :". Powered by SolveForX AI math tutor.
Step-by-Step Solution
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 21log(9)−21log(1).
Step 4: Simplify the expression
Since log1=0, the expression simplifies to 21log9. We can also write 21log9 as log(91/2) which is log3.
Step 5: Find the value of k
We are given that the integral equals logk. By comparing our result log3 with logk, we find that k=3.