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Integrate ln (sin x) from 0 to π2.

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=0π2ln(sinx)dx …(i)
I=0π2ln[sin(π2x)]dx [Using property of definite integrals]
[Property of definite integral: In a definite integral, 0af(x)dx=0af(xa)dx]
I=0π2ln(cosx)dx …(ii) [since, sin (π2x) = cos x]
Adding equations (i) and (ii), we get
2I=0π2ln(sinx)dx+0π2ln(cosx)dx
2I=0π2[ln(sinx)+ln(cosx)]dx [Using property of integrals]
[Property of definite integrals: 0π2f(x)dx+0π2g(x)dx=0π2[f(x)+g(x)]dx]
2I=0π2ln(sinx×cosx)dx
[Property of logarithm: ln (a) +ln (b) = ln (ab)]
2I=0π2ln[2sinx×cosx2]dx
2I=0π2ln[sin2x2]dx [Since, 2 × sinx × cosx = sin 2x]
[Applying logarithm property: log(ab)=logalogb]
2I=0π2[ln(sin2x)ln2]dx
2I=0π2lnsin2xdx+0π2ln2dx …(iii)
Assume, I1=0π2lnsin2xdx
Putting 2x = y
For x = 0, y = 0
For x=π2,y=π
Differentiating both sides of (2x = y) with respect to x, we get
2=dydxdx=dy2
Now, I1=0π2ln(siny)dy2=120π2ln(siny)dy
I1=12I [Since, abf(x)dx=abf(y)dy]
Putting value of I1in equation (iii), we have
2I=I2+ln2[x]0π2 [0π2ln2dx=ln2[x]0π2]
2II2=π2ln2
Separating I and constant parts
3I2=π2ln2
Multiplying both sides by 2
3I=πln2
I=π3ln2

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.