Mute said:Those are indefinite integrals, so they're only unique up to an additive constant. Notice:
[tex]\frac{1}{2}\log(2x+2) = \frac{1}{2}\log(x+1) + \log\sqrt{2}[/tex]
hence the two integrals differ only by a constant, and both are correct.
Logarithmic functions are mathematical functions that are the inverse of exponential functions. They involve the use of a logarithm, which is the power to which a base number must be raised to equal a given value. In other words, logarithmic functions tell you what power you need to raise a base number to in order to get a certain value.
To integrate logarithmic functions, you can use the formula ∫ln(x)dx = xln(x) - x + C, where C is the constant of integration. You can also use the substitution method, where you substitute u = ln(x) and du = 1/x dx, and then integrate using the formula ∫u du = u^2/2 + C.
Logarithmic functions are important because they are used to model various real-world phenomena, such as population growth, radioactive decay, and sound intensity. They are also used in many fields of science, including physics, chemistry, and biology, to solve complex problems.
The main difference between natural logarithmic functions and common logarithmic functions is the base number. Natural logarithmic functions have a base of e (approximately 2.718), while common logarithmic functions have a base of 10. This means that the inverse of a natural logarithm is an exponential function with base e, while the inverse of a common logarithm is an exponential function with base 10.
To use logarithmic functions to solve real-world problems, you can first identify the phenomenon being modeled and the given information. Then, you can set up an equation using the appropriate logarithmic function and solve for the unknown variable. It is important to remember to check your answer in the original equation to ensure it is valid.