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

Finding the Area Bounded by a Curve and the xaxis:
 Use the formula,
Area = ∫[a,b] f(x) dx
, wherea
andb
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.
 Use the formula,

Finding the Volume of a Solid Revolution:
 Use the formula,
Volume = ∫[a,b] πf(x)^2 dx
, wherea
andb
are the limits of integration, f(x) is the function representing the curve being revolved, andπ
is the constant pi (approximately 3.14).  Alternatively, you can use the
Disk Method
orShell Method
to find the volume.
 Use the formula,

Evaluating Improper Integrals:
 If the function f(x) has an infinite discontinuity at
a
orb
, 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.
 If the function f(x) has an infinite discontinuity at

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

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.

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.

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.

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:

Finding the Area Bounded by a Curve and the xaxis:
 Follow the same approach as in JEE Main, using the formula
Area = ∫[a,b] f(x) dx
.
 Follow the same approach as in JEE Main, using the formula

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
, theDisk Method
, or theShell Method
.
 Follow the same approach as in JEE Main, using the formula

Using Definite Integrals to Evaluate Improper Integrals:
 Follow the same approach as in JEE Main, using the limit definition of the definite integral.