$\text { Determine whether the improper integral } \int _ { 0 } ^ { \infty } \frac { e ^ { 2 x } } { 1 + e ^ { 4 x } } d x \text { diverges or converges. }$ Evaluate the integral if it converges.
A) diverges
B) converges

Question

$\text { Determine whether the improper integral } \int _ { 0 } ^ { \infty } \frac { e ^ { 2 x } } { 1 + e ^ { 4 x } } d x \text { diverges or converges. }$ Evaluate the integral if it converges. 
A) diverges
B) converges

Quizplus · Accepted Answer

We can use the comparison test. Note that for all $x \geq 0$, $e^{4x} > 1$ so we have
$$\frac{e^{2x}}{1+e^{4x}} < \frac{e^{2x}}{e^{4x}} = e^{-2x}.$$
Since $\int_{0}^{\infty} e^{-2x} dx$ converges, we conclude that $\int_{0}^{\infty} \frac{e^{2x}}{1+e^{4x}} dx$ converges by comparison. To evaluate the integral, we use the substitution $u = e^{2x}$, $du = 2e^{2x} dx$. Then
$$\int_{0}^{\infty} \frac{e^{2x}}{1+e^{4x}} dx = \frac{1}{2} \int_{1}^{\infty} \frac{du}{u^2+1} = \frac{1}{2} \cdot \tan^{-1}(u) \bigg\vert_{1}^{\infty} = \frac{\pi}{8}.$$

Evaluate the Integral If It Converges