Question

The integral \[ \int \frac{1 + x \cos x}{x \left[ 1 - x^2 \left( e^{\sin x} \right)^2 \right]} \, dx \] Options:

• A. \( \frac{1}{2} \log \left| \frac{(xe^{\sin x})^2}{(xe^{\sin x})^2 + 1} \right| + c \)
• B. \( -\frac{1}{2} \log \left| \frac{(xe^{\sin x})^2}{(xe^{\sin x})^2 + 1} \right| + c \)
• C. \( \frac{1}{2} \log \left| \frac{(xe^{\sin x})^2}{(xe^{\sin x})^2 - 1} \right| + c \)
• D. \( -\frac{1}{2} \log \left| \frac{(xe^{\sin x})^2}{(xe^{\sin x})^2 - 1} \right| + c \)

Hint:

When solving integrals involving exponential and trigonometric functions, recognize patterns and apply standard substitution or logarithmic integration techniques to simplify the problem.

Answer 1

The Correct Option is C

We are tasked with solving the integral: \[ I = \int \frac{1 + x \cos x}{x \left[ 1 - x^2 \left( e^{\sin x} \right)^2 \right]} \, dx \] Step 1: Simplify the integrand.
The expression can be simplified by recognizing the pattern in the denominator. The key part of the denominator suggests the use of a trigonometric substitution or simplifying the exponential terms to make the integral more manageable. \[ I = \int \frac{1 + x \cos x}{x \left[ 1 - x^2 \left( e^{\sin x} \right)^2 \right]} \, dx \] Step 2: Apply integration technique.
To solve this, we use standard integration techniques that involve simplifying logarithmic integrals and using properties of exponents. The integral results in the following expression: \[ \frac{1}{2} \log \left| \frac{(xe^{\sin x})^2}{(xe^{\sin x})^2 - 1} \right| + c \] Final Answer: \[ \boxed{\frac{1}{2} \log \left| \frac{(xe^{\sin x})^2}{(xe^{\sin x})^2 - 1} \right| + c} \]