yaho8888 said:Homework Statement
intrgral of 6ln(x^2-9)
solution
6((x+3)ln(x+3)-(x+3)+(x-3)ln(x-3)+ln(x-3))+c
check if it is right?
yaho8888 said:Why is ln(x-3)?
The concept behind evaluating the integral of 6ln(x^2-9) involves finding the area under the curve of the function 6ln(x^2-9) between two given points. This is done by finding the anti-derivative of the function and plugging in the values of the given points into the anti-derivative to get the final value.
To find the anti-derivative of 6ln(x^2-9), you can use the formula for integration by parts or integration by substitution. Both methods involve breaking down the function into smaller, simpler parts and using known integration rules to find the anti-derivative.
The integral of 6ln(x^2-9) is undefined when the function itself is undefined. Since ln(x) is only defined for positive values of x, the integral of 6ln(x^2-9) is undefined when x^2-9 is less than or equal to 0. This means that the possible values of x for which the integral is undefined are x=3 and x=-3.
Yes, the integral of 6ln(x^2-9) can be evaluated using numerical methods such as the trapezoidal rule, Simpson's rule, or the midpoint rule. These methods involve approximating the area under the curve by dividing it into smaller trapezoids or rectangles and using their areas to estimate the integral.
The integral of 6ln(x^2-9) has various applications in physics, engineering, and economics. For example, it can be used to calculate the work done by a variable force, the displacement of a particle under varying acceleration, or the average cost of production for a certain product. It is also used in the study of logarithmic functions and their derivatives.