‹ Back to the paper
Solve the following.
Evaluate: ∫_(-π/2)^(π/2) (cos x)/(1 + eˣ) dx
Evaluate: $\int_{-\pi/2}^{\pi/2} \frac{\cos x}{1 + e^x} dx$
Answer
Answer
Integral: $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(\frac{\cos x}{1+e^x}\right) dx$
Let $I = \int_{-\pi/2}^{\pi/2}\frac{\cos x}{1+e^x}dx$.
Using $\int_a^b f(x)dx = \int_a^b f(a+b-x)dx$ with $a + b = 0$: $I = \int_{-\pi/2}^{\pi/2}\frac{\cos x}{1+e^{-x}}dx = \int_{-\pi/2}^{\pi/2}\frac{e^x\cos x}{1+e^x}dx$.
Adding: $2I = \int_{-\pi/2}^{\pi/2}\cos x\,dx = \left[\sin x\right]_{-\pi/2}^{\pi/2} = 1 - (-1) = 2$.
Hence $I = 1$.
Final answer: 1