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definite integral using property
integral of ln(sin x) from 0  to pi/2

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integral of ln(sin x) from 0  to pi/2
Take the given integral as I
apply the property

of definite integrals to change x to [(pi/2) - x]
use the property of trigonometry to change
sin
[(pi/2) - x] to cosx
add the two values of I
apply the property of logarithm to make into
a single expression and use trigonometry formula for sin2x
and use the property of log to split into two integrals and
evaluate the simple integral involving ln(2)

now use the substitution u = 2x in the other integral
and change the limits  in terms of u. [0 to pi]

Use the property  of definite integral to
change back the limits to 0  to pi/2.

You will get back the original integral I
 
Now solve for I.

integration formula

integral of ln(sin x) from 0 to pi/2
integral of ln(sin x) from 0 to pi/2

some problems from integral calculus

* Integrate e^{x^(1/3)}dx

* evaluate integral of [e^(-x) +1] ^2

*evaluate ∫ (sin 3x)^4 dx with limits 0 to pi

* evaluate the integral of sqrt(tan  x)

* evaluate integral of { sqrt{1+(x^2)} } / {x^2}
using integration by parts


*∫ sec³x dx using integration by parts

*integral of (e^(ax) )  cosbx using integration by parts


Area using integral

* evaluate the area of an ellipse (x² /a²) + (y²/b²) =1 using integration

* find the area between y² = x and x² =y

* find the area bounded b y =x² and y = 3x

other questions and problems:


*integral calculus integration formula





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trigonometric identity and ratio of  certain standard angles

*list of differentiation formula     *integral calculus integration formula
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