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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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