Question

Integrate the function: $\frac{2+sin2x}{1+cos2x}e^{x}$

Answer 1

$I=\int(\frac{2+2sinx}{1+cos2x})e^{x}$

$=\int (\frac{2+2sinxcosx}{2cos^{2}x})e^{x}$

$=\int(\frac{1+sinxcosx}{cos^{2}x})e^{x}$

$=∫(sec^{2}x+tanx)e^{x}$

$Let ƒ(x)=tanx⇒ƒ'(x)=sec^{2}x$

$∴I=∫(ƒ(x)+ƒ'(x)]e^{x}dx$

$=e^{x}ƒ(x)+C$

$=e^{x}tanx+C$