If there four options are given to a Defenite Integration then what shortcut trick we should choose to get the correct answer.
Like in Indefinte Integration we find the differtation of the answer to reach the function given to integrate.
You can try to eliminate some answers as follows.
|int f(x)dx| <= int |f(x)|dx <=
max[f(x)]*(length on the integration interval).
Options that are inconsistent with this inequality can be eliminated.
To find the shortcut trick for solving a definite integration problem with four given options, you can use the inequality:
|int f(x)dx| ≤ int |f(x)|dx ≤ max[f(x)] * (length of the integration interval).
Here's how you can use this trick to eliminate some options:
1. Evaluate the definite integral of |f(x)|dx for each option using any integration technique or method. This will give you the absolute value of the area under the curve for each option.
2. Calculate the maximum value of f(x) within the given integration interval. This can be done by finding the critical points of f(x) and evaluating the function at those points, as well as at the endpoints of the interval.
3. Multiply the maximum value of f(x) by the length of the integration interval. This will give you the maximum possible value of the integral.
4. Compare each option's absolute value of the integral (calculated in step 1) with the maximum possible value of the integral (calculated in step 3).
5. Any option that violates the inequality |int f(x)dx| ≤ int |f(x)|dx ≤ max[f(x)] * (length of the integration interval) can be eliminated. In other words, if an option's absolute value of the integral is greater than the maximum possible value, or it is less than the minimum possible value (which is 0), then that option can be ruled out.
By using this shortcut trick, you can eliminate options that do not satisfy the inequality and narrow down the choices to the correct answer.