Evaluate: $\int \frac{1+\sin x}{1+\cos x}\cdot e^x\,dx$

Evaluate: ∫ (1+sin x)/(1+cos x)· exdx

3.3 · Q141

\int \frac{1+\sin x}{1+\cos x}\cdot e^x\,dx

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Use $1+\sin x=\left(\cos\frac x2+\sin\frac x2\right)^2$ and $1+\cos x=2\cos^2\frac x2$ :

$\dfrac{1+\sin x}{1+\cos x}=\dfrac{\left(\cos\frac x2+\sin\frac x2\right)^2}{2\cos^2\frac x2}=\dfrac12\left(1+\tan\dfrac x2\right)^2=\dfrac12\left(1+2\tan\dfrac x2+\tan^2\dfrac x2\right)$ …

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Q.Evaluate: $\int \frac{1+\sin x}{1+\cos x}\cdot e^x\,dx$

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Q.Evaluate: ∫1+sin⁡x1+cos⁡x⋅ex dx\int \frac{1+\sin x}{1+\cos x}\cdot e^x\,dx∫1+cosx1+sinx​⋅exdx

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Q.Evaluate: $\int \frac{1+\sin x}{1+\cos x}\cdot e^x\,dx$

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