Question

The value of \(\int_{0}^{\sqrt{\ln\left(\frac{\pi}{2}\right)}} \cos\left(e^{x^2}\right)\, 2x e^{x^2}\, dx\) is

• A. 1
• B. \(1+\sin 1\)
• C. \(1-\sin 1\)
• D. \(\sin 1 -1\)

Hint:

Look for pattern \(2x e^{x^2}dx\) → substitute \(t=e^{x^2}\).

Answer 1

The Correct Option is C

Step 1: Substitute.
Let: \[ t = e^{x^2} \Rightarrow dt = 2x e^{x^2} dx \]

Step 2: Change limits.
\[ x=0 \Rightarrow t=1 \] \[ x=\sqrt{\ln(\pi/2)} \Rightarrow t = \frac{\pi}{2} \]

Step 3: Transform integral.
\[ \int_1^{\pi/2} \cos t \, dt \] \[ = \sin t \Big|_1^{\pi/2} = \sin\left(\frac{\pi}{2}\right) - \sin(1) \] \[ = 1 - \sin 1 \]