Shortcuts and Tricks to Solve JEE Main, Advanced, and CBSE Class 12 Definite Integral Problems

JEE Main

1. Finding the Area Bounded by a Curve and the x-axis:

   • Use the formula, Area = ∫[a,b] f(x) dx, where a and b are the limits of integration and f(x) is the function representing the curve.
   • Split the area into simple regions and integrate each region separately.

2. Finding the Volume of a Solid Revolution:

   • Use the formula, Volume = ∫[a,b] πf(x)^2 dx, where a and b are the limits of integration, f(x) is the function representing the curve being revolved, and π is the constant pi (approximately 3.14).
   • Alternatively, you can use the Disk Method or Shell Method to find the volume.

3. Evaluating Improper Integrals:

   • If the function f(x) has an infinite discontinuity at a or b, or if the limit of integration is infinite, then the integral is considered improper.
   • Use the limit definition of the definite integral to evaluate improper integrals.

4. Applying Definite Integrals to Problems in Physics and Engineering:

   • Use the concept of work done by a variable force to find the work done by a force over a certain distance.
   • Use the concept of the center of mass to find the center of mass of a given object or system.

JEE Advanced

1. Definite Integrals Involving Trigonometric Functions:

   • Use trigonometric identities to simplify the integral and then integrate using standard formulas.
   • Use the definite integral properties to evaluate integrals involving trigonometric functions.

2. Definite Integrals Involving Inverse Trigonometric Functions:

   • Use the substitution method to transform the integral into an integral involving trigonometric functions.
   • Then, integrate using standard formulas.

3. Definite Integrals Involving Logarithmic Functions:

   • Use the substitution method to transform the integral into an integral involving natural logarithms (ln).
   • Then, integrate using the power rule of integration.

4. Definite Integrals Involving Exponential Functions:

   • Use the substitution method to transform the integral into an integral involving natural logarithms (ln).
   • Then, integrate using integration by parts.

CBSE Class 12:

1. Finding the Area Bounded by a Curve and the x-axis:

   • Follow the same approach as in JEE Main, using the formula Area = ∫[a,b] f(x) dx.

2. Finding the Volume of a Solid Revolution:

   • Follow the same approach as in JEE Main, using the formula Volume = ∫[a,b] πf(x)^2 dx, the Disk Method, or the Shell Method.

3. Using Definite Integrals to Evaluate Improper Integrals:

   • Follow the same approach as in JEE Main, using the limit definition of the definite integral.