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Pattern - 16 of Indefinite Integration

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⋆ Pattern – 16

∫ $e^{ax+b} [ a f(x) + f' (x) ] dx = e^{ax+b} f(x) + c$

1) $\ I=\ \int e^{2x}\ \left(\frac{1+\sin{2x}}{1+\cos{2x}}\right)\ dx$

Solution.

Here,

I = $\ \int{e^{2x}\ \left(\frac{1+2\sin{x\ }\times \cos{x}}{2\cos^2{x}}\right)}\ dx$

$=\frac{1}{2}\ \int e^{2x}\left(\sec^2{x}+2\tan{x}\right)\ dx$

$=\frac{1}{2}\ \int{e^{2x}\ \left(2\tan{x}+\sec^2{x}\right)}\ dx$

$=\frac{1}{2}\ e^{2x}\tan{x}+c$