Patterns of problems
7 min read

Integrals

- Understand the patterns for better solving
1
Pattern: Integrals of standard functions
Description: Most of the problems on finding integrals of standard functions are of direct formula based. So you have to remember the basic integrals like integral of , trigonometric functions etc. Questions in this pattern are based on finding integrals of standard functions. For example of finding integral of trigonometric functions, polynomial functions or sum/ difference of standard functions. Remember to apply the property =
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2
Pattern: Integrals of trigonometric functions. Description: While finding integrals of trigonometric functions for standard functions like sin, cos, tan and their reciprocals, sum or differences of these standard functions. We can directly use the result. But in case of integrals other than standard integrals. Use identities that simplify to standard trigonometric functions so that the integration can be done easily. Questions can be asked based on finding integration which is a combination of different trigonometric functions.
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3
Pattern: Integrals of functions (type u.v) by parts
Description: While finding integrals of the type of functions , we have to use by integration by parts formula = . The function needs to be selected based on order Inverse trigonometric, logarithmic, algebraic, trigonometric and exponential functions (ILATE).Questions can be asked based on finding integrals of , and being combination of inverse trigonometric, logarithmic, algebraic, trigonometric and exponential functions.
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4
Pattern: Integrals of rational functions
Description: While integrating functions of the type . Express as partial fractions. Find the values of constants and then integrate. Questions can be asked based on integrating polynomial functions which have repeated linear factors, non- repeated linear factors, repeated and non repeated quadratic equations.
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5
Pattern: Integrals of product or ratio of linear and irrational linear functions
Description: While integrating linear functions of the type and , write substitute t in place of and rewrite in terms of t. Replace the functions, dx in terms of t. Then we find the integral. Questions can be asked based on integrating functions of the type , and where , and are linear functions of x
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6
Pattern: Finding definite Integrals by splitting limits
Description: While evaluating integrals using definite integration, the value of functions changes in the given intervals. So we use the property of Integrals by splitting limits so that value of the functions remains same in that split limit. Questions can be asked based finding the definite integral of modulus functions, step functions, trigonometric function with limits where the value of the function changes.
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7
Pattern: Integrals of even and odd functions
Description: While finding integrals of the type , the result depends on whether f(x) is odd or even function.
if is even function
if is odd function. Questions can be asked based on trigonometric functions, polynomial functions.
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