Trigonometric Integrals

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In a physics problem modeling wave interference, the intensity pattern is given by the integral involving powers of sine and cosine. Determine the indefinite integral: $\int \sin^4(3x) \cos^2(3x) \, dx$ .

\frac{x}{16} - \frac{\sin(12x)}{96} - \frac{\sin^3(6x)}{144} + C

\frac{x}{16} - \frac{\sin(6x)}{48} - \frac{\sin(12x)}{192} + \frac{\sin^3(6x)}{72} + C

\frac{x}{16} - \frac{\sin(12x)}{192} - \frac{\sin^3(6x)}{144} + C

\frac{3x}{16} - \frac{\sin(12x)}{64} - \frac{\sin^3(6x)}{48} + C

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Determine the antiderivative: $\int \sin^2 x \cos^3 x \, dx$ .

\frac{1}{4}\cos^4 x - \frac{1}{6}\cos^6 x + C

\frac{1}{3}\sin^3 x - \frac{1}{5}\sin^5 x + C

\frac{1}{8}x - \frac{1}{32}\sin 4x + C

\frac{1}{3}\sin^3 x - \frac{1}{3}\sin^5 x + C

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Evaluate the integral $\int \sin^3 x \cos^2 x \, dx$ , which arises in the analysis of rotational motion.

\frac{1}{3}\cos^3 x - \frac{1}{5}\cos^5 x + C

\frac{1}{3}\sin^3 x - \frac{1}{5}\sin^5 x + C

\frac{1}{5}\cos^5 x - \frac{1}{3}\cos^3 x + C

\frac{1}{3}\cos^3 x - \frac{1}{2}\cos^5 x + C

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Trigonometric Integrals

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