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Question #41502

Difficulty: 8/10 | 🟢 Free

The Question

Let $L_1$ and $L_2$ be two variable lines in 3D space with direction vectors $\vec{d_1} = \sin x \hat{i} + \cos x \hat{j} + \hat{k}$ and $\vec{d_2} = \cos x \hat{i} + \sin x \hat{j} + \hat{k}$ respectively. Evaluate the integral $I = \int_{0}^{\pi} |\vec{d_1} \times \vec{d_2}| \, dx$

Options

A.
$2\sqrt{2} + 2\ln(\sqrt{2}+1)$
B.
$\sqrt{3} + \ln(2+\sqrt{3})$
C.
$2\sqrt{2} + \ln(\sqrt{2}+1)$
D.
$2\sqrt{2} - \ln(\sqrt{2}+1)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Trigonometric Identical Substitutions|Lagrange Identity|Angle between two lines

Why This Question Is Tricky

Common Mistakes

This question tests the application of Lagrange's Identity to simplify vector magnitudes into integrable trigonometric forms, combined with a classic definite integral trap.

Students often compute the cross product magnitude but forget that $\sqrt{(\cos x - \sin x)^2}$ must be evaluated as $|\cos x - \sin x|$ .

In the context of directional vectors, the magnitude is strictly non-negative, forcing a domain split at $x = \pi/4$ where the relative orientation of the lines shifts.

Ignoring this absolute value leads to a direct integration from $0$ to $\pi$ , resulting in an incorrect scalar value, completely missing the geometric reality of the area under a magnitude curve.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Trigonometric Identical Substitutions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free