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4. Indefinite Integrals 
4.1 Does the integral ?
-?? ?? ?v
?? +?? ?? -?? ???? exist? If so, find its value. 
(2010 : 12 Marks) 
Solution: 
Given the integral is ?
-1
1
?v
1+?? 1-?? ???? 
Rationalizing integrand, we get 
                                                ? ?
1
-1
?
v(1+?? )
2
v1-?? 2
???? =? ?
1
-1
?
1+?? v1-?? 2
????
 Let ?? -sin?? ,?                                            ???? =cos ?????? ? Integral becomes ? ?
?? -?? ?
1+sin ?? v1-sin
2
 ?? ???? ×cos ??                                                                                  =? ?
?? /2
-?? /2
?
1+sin ?? |cos ?? |
×cos ?????? =? ?
?? /2
-?? /2
?(1+sin ?? )????
                                                                                  =[?? ]
-?? /2
?? /2
+0=?? 
 
 
 
 
 
 
 
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FAQs on Indefinite Integrals - Mathematics Optional Notes for UPSC

1. What is an indefinite integral in UPSC mathematics?
Ans. An indefinite integral in UPSC mathematics is the reverse process of differentiation. It involves finding a function that, when differentiated, gives the original function.
2. How is the symbol for indefinite integral represented in UPSC mathematics?
Ans. The symbol for indefinite integral in UPSC mathematics is ∫ (integral symbol), followed by the function to be integrated and dx to represent the variable of integration.
3. What is the significance of finding indefinite integrals in UPSC mathematics?
Ans. Finding indefinite integrals in UPSC mathematics helps in solving problems related to areas under curves, velocity, distance, and various other physical and mathematical concepts.
4. Can indefinite integrals have multiple solutions in UPSC mathematics?
Ans. Yes, indefinite integrals can have multiple solutions in UPSC mathematics due to the presence of arbitrary constants that arise during the integration process.
5. How can one evaluate indefinite integrals efficiently in UPSC mathematics?
Ans. One can evaluate indefinite integrals efficiently in UPSC mathematics by using integration techniques such as substitution, integration by parts, trigonometric integrals, and partial fractions.
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