Question
Integrate the following functions w.r.t. x:
$\frac{e^x \cdot \log \left(\sin e^x\right)}{\tan \left(e^x\right)}$
$\frac{e^x \cdot \log \left(\sin e^x\right)}{\tan \left(e^x\right)}$
$=\int \log \left(\sin e^x\right) \cdot e^x \cot \left(e^x\right) d x$
Put $\log \left(\sin e^x\right)=t \quad \therefore \frac{1}{\sin \left(e^x\right)} \cdot \cos \left(e^x\right) \cdot e^x d x=d t$
$\therefore e^x \cdot \cot \left(e^x\right) d x=d t$
$\begin{aligned} \therefore I & =\int t d t=\frac{t^2}{2}+c \\ & =\frac{1}{2}\left[\log \left(\sin e^x\right)\right]^2+c .\end{aligned}$
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$(\bar{a}+\bar{b}+\bar{c}) \times \bar{c}+(\bar{a}+\bar{b}+\bar{c}) \times \bar{b}+(\bar{b}+\bar{c}) \times \bar{a}=2 \bar{a} \times \bar{c}$
Question is modified.
For any vectors $\bar{a}, \bar{b}, \bar{c}$ show that
$(\bar{a}+\bar{b}+\bar{c}) \times \bar{c}+(\bar{a}+\bar{b}+\bar{c}) \times \bar{b}+(\bar{b}-\bar{c}) \times \bar{a}$
$=2 \bar{a} \times \bar{c}$.