1. \( \int \operatorname{sen} 2 x \operatorname{sen} 3 x d x \) 2. \( \int \operatorname{sen} x \cos 3 x d x \)

Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \sin\left(2x\right)\sin\left(3x\right) dx\) - step1: Evaluate the integral: \(\frac{\sin\left(\left(2-3\right)x\right)}{2\left(2-3\right)}-\frac{\sin\left(\left(2+3\right)x\right)}{2\left(2+3\right)}\) - step2: Subtract the numbers: \(\frac{\sin\left(-x\right)}{2\left(2-3\right)}-\frac{\sin\left(\left(2+3\right)x\right)}{2\left(2+3\right)}\) - step3: Add the numbers: \(\frac{\sin\left(-x\right)}{2\left(2-3\right)}-\frac{\sin\left(5x\right)}{2\left(2+3\right)}\) - step4: Subtract the numbers: \(\frac{\sin\left(-x\right)}{2\left(-1\right)}-\frac{\sin\left(5x\right)}{2\left(2+3\right)}\) - step5: Add the numbers: \(\frac{\sin\left(-x\right)}{2\left(-1\right)}-\frac{\sin\left(5x\right)}{2\times 5}\) - step6: Transform the expression: \(\frac{-\sin\left(x\right)}{2\left(-1\right)}-\frac{\sin\left(5x\right)}{2\times 5}\) - step7: Simplify: \(\frac{-\sin\left(x\right)}{-2}-\frac{\sin\left(5x\right)}{2\times 5}\) - step8: Multiply the numbers: \(\frac{-\sin\left(x\right)}{-2}-\frac{\sin\left(5x\right)}{10}\) - step9: Simplify the expression: \(\frac{1}{2}\sin\left(x\right)-\frac{\sin\left(5x\right)}{10}\) - step10: Simplify the expression: \(\frac{1}{2}\sin\left(x\right)-\frac{1}{10}\sin\left(5x\right)\) - step11: Add the constant of integral C: \(\frac{1}{2}\sin\left(x\right)-\frac{1}{10}\sin\left(5x\right) + C, C \in \mathbb{R}\) Calculate the integral \( \int \sin(x) \cos(3 x) \, dx \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \sin\left(x\right)\cos\left(3x\right) dx\) - step1: Evaluate the integral: \(-\frac{\cos\left(\left(1+3\right)x\right)}{2\left(1+3\right)}-\frac{\cos\left(\left(1-3\right)x\right)}{2\left(1-3\right)}\) - step2: Add the numbers: \(-\frac{\cos\left(4x\right)}{2\left(1+3\right)}-\frac{\cos\left(\left(1-3\right)x\right)}{2\left(1-3\right)}\) - step3: Subtract the numbers: \(-\frac{\cos\left(4x\right)}{2\left(1+3\right)}-\frac{\cos\left(-2x\right)}{2\left(1-3\right)}\) - step4: Add the numbers: \(-\frac{\cos\left(4x\right)}{2\times 4}-\frac{\cos\left(-2x\right)}{2\left(1-3\right)}\) - step5: Subtract the numbers: \(-\frac{\cos\left(4x\right)}{2\times 4}-\frac{\cos\left(-2x\right)}{2\left(-2\right)}\) - step6: Multiply the numbers: \(-\frac{\cos\left(4x\right)}{8}-\frac{\cos\left(-2x\right)}{2\left(-2\right)}\) - step7: Transform the expression: \(-\frac{\cos\left(4x\right)}{8}-\frac{\cos\left(2x\right)}{2\left(-2\right)}\) - step8: Multiply the numbers: \(-\frac{\cos\left(4x\right)}{8}-\frac{\cos\left(2x\right)}{-4}\) - step9: Simplify the expression: \(-\frac{1}{8}\cos\left(4x\right)-\frac{\cos\left(2x\right)}{-4}\) - step10: Simplify the expression: \(-\frac{1}{8}\cos\left(4x\right)-\left(-\frac{1}{4}\cos\left(2x\right)\right)\) - step11: Calculate: \(-\frac{1}{8}\cos\left(4x\right)+\frac{1}{4}\cos\left(2x\right)\) - step12: Add the constant of integral C: \(-\frac{1}{8}\cos\left(4x\right)+\frac{1}{4}\cos\left(2x\right) + C, C \in \mathbb{R}\) Aquí están las soluciones a las integrales solicitadas: 1. Para la integral \( \int \sin(2x) \sin(3x) \, dx \): \[ \int \sin(2x) \sin(3x) \, dx = \frac{1}{2}\sin(x) - \frac{1}{10}\sin(5x) + C, \quad C \in \mathbb{R} \] 2. Para la integral \( \int \sin(x) \cos(3x) \, dx \): \[ \int \sin(x) \cos(3x) \, dx = -\frac{1}{8}\cos(4x) + \frac{1}{4}\cos(2x) + C, \quad C \in \mathbb{R} \] Si necesitas más ayuda o explicaciones sobre los pasos, no dudes en preguntar.