
Integrate ln (sin x) from 0 to
Answer
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Hint:Here, use properties of definite integrals to convert sin x to cos x and add the two integrals. Using properties of logarithm, simplify the integral. Also using properties of trigonometric functions simplify the integral to get the result
Complete step-by-step answer:
Let …(i)
[Using property of definite integrals]
[Property of definite integral: In a definite integral, ]
…(ii) [since, sin = cos x]
Adding equations (i) and (ii), we get
[Using property of integrals]
[Property of definite integrals: ]
[Property of logarithm: ln (a) +ln (b) = ln (ab)]
[Since, 2 × sinx × cosx = sin 2x]
[Applying logarithm property: ]
…(iii)
Assume,
Putting 2x = y
For x = 0, y = 0
For
Differentiating both sides of (2x = y) with respect to x, we get
Now,
[Since, ]
Putting value of in equation (iii), we have
Separating I and constant parts
Multiplying both sides by 2
Note:In these types of questions, always use properties of definite integral do not go for actual integration steps. Try to use trigonometric identities, properties of definite integrals and logarithmic properties to simplify the given problem. Always careful while doing the steps and using properties. If we solve these types of integral problems directly it becomes much more complicated, and it may happen that you will not get the final results. These properties can be used only for definite integrals not for indefinite integrals. For doing these types of problems brush up trigonometric identities, and should also be aware about logarithmic basic properties, without which you will not be able to solve the problem.
Complete step-by-step answer:
Let
[Property of definite integral: In a definite integral,
Adding equations (i) and (ii), we get
[Property of definite integrals:
[Property of logarithm: ln (a) +ln (b) = ln (ab)]
[Applying logarithm property:
Assume,
Putting 2x = y
For x = 0, y = 0
For
Differentiating both sides of (2x = y) with respect to x, we get
Now,
Putting value of
Separating I and constant parts
Multiplying both sides by 2
Note:In these types of questions, always use properties of definite integral do not go for actual integration steps. Try to use trigonometric identities, properties of definite integrals and logarithmic properties to simplify the given problem. Always careful while doing the steps and using properties. If we solve these types of integral problems directly it becomes much more complicated, and it may happen that you will not get the final results. These properties can be used only for definite integrals not for indefinite integrals. For doing these types of problems brush up trigonometric identities, and should also be aware about logarithmic basic properties, without which you will not be able to solve the problem.
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