Question

Evaluate: \[ \int_{0}^{\pi/4} e^{\tan^2 \theta} \sin^2 \theta \tan \theta \, d\theta = \]

• A. \( \frac{1}{2}\left(\frac{e}{2} - 1\right) \)
• B. \( \frac{e}{2} - 1 \)
• C. \( \frac{\pi}{2} \)
• D. \( 2\left(\frac{\pi}{2} - e\right) \)

Hint:

Use substitution techniques and express all trig functions in terms of tangent for simplification.

Answer 1

The Correct Option is A

Let \( t = \tan \theta \Rightarrow dt = \sec^2 \theta d\theta \), and \( \sin^2 \theta = \frac{t^2}{1 + t^2} \), so \[ \int e^{t^2} \cdot \frac{t^3}{(1 + t^2)^2} dt \] After substitution and limits change from \( t = 0 \) to \( t = 1 \), solve to get the answer.