Integral of exponential function

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$\int \left(\sin2x\; \cos2x\right)\;e^{{\sin}^22x} \; dx$

**Answer**

$\int \left(\sin2x\; \cos2x\right)\;e^{{\sin}^22x} \; dx$

Substitute $u= {\sin}^22x$. So, $du =4\sin2x\;\cos2x\; dx$

And, $\frac{1}{4}\; du = \sin2x\; \cos 2x\; dx$

Then $\int \left(\sin2x\; \cos2x\right)e^{{\sin}^22x} \; dx$

$=\frac{1}{4}\int e^{u} \; du$

$=\frac{1}{4}e^u + C $

Substituting back $u= {\sin}^22x$

$=\frac{1}{4}e^{{\sin}^2 2x} + C$