Please explain how it is (ln 4)
¡Ò^8 2 x^-1 dx= (ln 8) ¨C (ln 2)
=(ln 4)
It's not possible to see the problem because the symbols are incomprehensible. Please retype the equation so that we can help you.
ln (8) - ln (2) = ln (4)
How is this possible
one of the most basic log rules is
Log(A/B) = LogA - LogB, as long as the base of the log is the same
since 8/2 = 4 it fits the pattern
you might also try to verify it on your calculator.
To find the value of the integral ∫ln(4) 2x^(-1) dx, we can use the formula for integrating ln(x), which is ∫ln(x) dx = x(ln(x) - 1) + C.
Let's break down the integral step by step:
1. We start with the integral ∫ln(4) 2x^(-1) dx.
2. We can rewrite 2x^(-1) as 2/x. So, the integral becomes ∫ln(4) 2/x dx.
3. Using the formula for integrating ln(x), we have ∫ln(4) 2/x dx = 2(ln|x| - 1) + C.
4. Remember that ln|x| represents the natural logarithm of the absolute value of x. In this case, since x is positive (as it is in the domain of ln(4)), we can remove the absolute value signs.
5. So, the integral simplifies to 2(lnx - 1) + C.
6. Finally, we replace lnx with ln(4), as specified in the problem, and simplify: 2(ln(4) - 1) + C = 2ln(4) - 2 + C.
7. The constant C represents the constant of integration, which can be any real number.
8. Therefore, the final result is 2ln(4) - 2 + C, which can be written as (ln(8) - ln(2)) + C.
Notice that ln(4) is equal to ln(2^2), and using the logarithmic property ln(a^b) = b*ln(a), we get ln(2^2) = 2*ln(2) = ln(4).
9. Since C can be any real number, we can rename it as a new constant, D: (ln(8) - ln(2)) + D.
10. So, the value of the integral is (ln(8) - ln(2)) + D.