# prove that pie/2 ingrete0 logsinxdx=-pie/2log2

Here, we need to evaluate .

First, use the property ${\int }_{0}^{a}f\left(x\right)dx={\int }_{0}^{a}f\left(a-x\right)dx$. So,

Adding (1) and (2), we get
$2I={\int }_{0}^{\frac{\mathrm{\pi }}{2}}\mathrm{log}\mathrm{sin}x+\mathrm{log}\mathrm{cos}xdx$
Next, add and subtract log 2 from the above integral. So, we get

Further, use 2x = t, then $2dx=dt$. So, when
Thus,

Next, use the property

Further, changing the variable t to x. We get

Hence proved.

• 2
What are you looking for?