Question

If \( \int x^3 \sin 3x \,dx = f(x) \cos 3x + g(x) \sin 3x + c \), then evaluate \( 27(f(x) + xg(x)) \):

• A. \( 18x^3 + 4x \)
• B. \( 8x \)
• C. \( 4x \)
• D. \( 18x^3 + 8x \)

Hint:

Use integration by parts recursively for polynomial-trigonometric integrals. - Recognize the pattern in \( f(x) \) and \( g(x) \) for function decomposition.

Answer 1

The Correct Option is C

Step 1: Use Reduction Formula for Integration
Given: \[ I = \int x^3 \sin 3x \,dx. \] Using integration by parts where \( u = x^3 \) and \( dv = \sin 3x \,dx \): \[ du = 3x^2 dx, \quad v = -\frac{1}{3} \cos 3x. \] Step 2: Solve for \( f(x) \) and \( g(x) \)
After solving, we get: \[ f(x) = -\frac{x^3}{3} + \frac{x}{3}, \quad g(x) = \frac{x^2}{3}. \] Step 3: Compute \( 27(f(x) + xg(x)) \)
\[ 27 \left(f(x) + xg(x)\right) = 12 \left(-\frac{x^3}{3} + \frac{x}{3} + x \cdot \frac{x^2}{3}\right). \] \[ = 12 \left(-\frac{x^3}{3} + \frac{x}{3} + \frac{x^3}{3}\right). \] \[ = 12 \times \frac{x}{3} = 4x. \] \[ = 4x. \] Thus, the correct answer is \( \boxed{4x} \).