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I have the equation: $$\frac{1}{\tau}\intop_{0}^{\tau}A\sin\left(\Omega t\right)\cdot A\sin\left(\Omega\left(t-\lambda\right)\right)\mathrm{d}t$$ for which the attempted solution is to convert the sine terms into complex natural exponents (engineering notation using j as imaginary unit) as $$\frac{A^{2}}{\tau}\intop_{0}^{\tau}\frac{e^{j\Omega t}-e^{-j\Omega t}}{2j}\cdot\frac{e^{j\Omega\left(t-\lambda\right)}-e^{-j\Omega\left(t-\lambda\right)}}{2j}\mathrm{d}t$$ the next step in the solution moves the $\frac{1}{2j}$ term outside of the integral to form $$\frac{-A^{2}}{4\tau}\intop_{0}^{\tau}\left(e^{j\Omega t}-e^{-j\Omega t}\right)\cdot\left(e^{j\Omega\left(t-\lambda\right)}-e^{-j\Omega\left(t-\lambda\right)}\right)\mathrm{d}t$$ I'm struggling to understand how $\frac{1}{2j}\rightarrow\frac{-1}{4}$ when being moved out of the integral.

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You're moving out $\dfrac1{2j}\cdot\dfrac1{2j}=-\dfrac14,$ not just $\dfrac1{2j}.$

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