Use the double-angle formulas to evaluate the following integrals. 97. \( \int_{0}^{\pi} \sin ^{2} x d x \) 98. \( \int_{0}^{\pi} \sin ^{4} x d x \) 99. \( \int \cos ^{2} 3 x d x \) 100. \( \int \sin ^{2} x \cos ^{2} x d x \)

Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int_{0}^{\pi } \sin^{2}\left(x\right) dx\) - step1: Evaluate the integral: \(\int \sin^{2}\left(x\right) dx\) - step2: Evaluate the integral: \(\frac{x}{2}-\frac{\sin\left(2x\right)}{4}\) - step3: Simplify the expression: \(\frac{x}{2}-\frac{1}{4}\sin\left(2x\right)\) - step4: Return the limits: \(\left(\frac{x}{2}-\frac{1}{4}\sin\left(2x\right)\right)\bigg |_{0}^{\pi }\) - step5: Calculate the value: \(\frac{\pi }{2}\) Calculate the integral \( \int \sin ^{2} x \cos ^{2} x \, dx \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \sin^{2}\left(x\right)\cos^{2}\left(x\right) dx\) - step1: Multiply the terms: \(\int \left(\sin\left(x\right)\cos\left(x\right)\right)^{2} dx\) - step2: Rewrite the expression: \(\int \frac{1}{4}\sin^{2}\left(2x\right) dx\) - step3: Use properties of integrals: \(\frac{1}{4}\times \int \sin^{2}\left(2x\right) dx\) - step4: Evaluate the integral: \(\frac{1}{4}\left(\frac{x}{2}-\frac{\sin\left(2\times 2x\right)}{4\times 2}\right)\) - step5: Simplify: \(\frac{1}{4}\left(\frac{x}{2}-\frac{1}{8}\sin\left(4x\right)\right)\) - step6: Use the distributive property: \(\frac{1}{4}\times \frac{x}{2}+\frac{1}{4}\left(-\frac{1}{8}\sin\left(4x\right)\right)\) - step7: Multiply the terms: \(\frac{x}{8}+\frac{1}{4}\left(-\frac{1}{8}\sin\left(4x\right)\right)\) - step8: Multiply the terms: \(\frac{x}{8}-\frac{1}{32}\sin\left(4x\right)\) - step9: Add the constant of integral C: \(\frac{x}{8}-\frac{1}{32}\sin\left(4x\right) + C, C \in \mathbb{R}\) Calculate the integral \( \int \cos ^{2} 3 x \, dx \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \cos^{2}\left(3x\right) dx\) - step1: Evaluate the integral: \(\frac{x}{2}+\frac{\sin\left(2\times 3x\right)}{4\times 3}\) - step2: Multiply the terms: \(\frac{x}{2}+\frac{\sin\left(6x\right)}{4\times 3}\) - step3: Multiply the numbers: \(\frac{x}{2}+\frac{\sin\left(6x\right)}{12}\) - step4: Simplify the expression: \(\frac{x}{2}+\frac{1}{12}\sin\left(6x\right)\) - step5: Add the constant of integral C: \(\frac{x}{2}+\frac{1}{12}\sin\left(6x\right) + C, C \in \mathbb{R}\) Calculate the integral \( \int_{0}^{\pi} \sin ^{4} x \, dx \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int_{0}^{\pi } \sin^{4}\left(x\right) dx\) - step1: Evaluate the integral: \(\int \sin^{4}\left(x\right) dx\) - step2: Evaluate the integral: \(-\frac{\sin^{4-1}\left(x\right)\cos\left(x\right)}{4}+\frac{4-1}{4}\times \int \sin^{4-2}\left(x\right) dx\) - step3: Subtract the numbers: \(-\frac{\sin^{3}\left(x\right)\cos\left(x\right)}{4}+\frac{4-1}{4}\times \int \sin^{4-2}\left(x\right) dx\) - step4: Subtract the numbers: \(-\frac{\sin^{3}\left(x\right)\cos\left(x\right)}{4}+\frac{4-1}{4}\times \int \sin^{2}\left(x\right) dx\) - step5: Subtract the numbers: \(-\frac{\sin^{3}\left(x\right)\cos\left(x\right)}{4}+\frac{3}{4}\times \int \sin^{2}\left(x\right) dx\) - step6: Simplify the expression: \(-\frac{1}{4}\sin^{3}\left(x\right)\cos\left(x\right)+\frac{3}{4}\times \int \sin^{2}\left(x\right) dx\) - step7: Evaluate the integral: \(-\frac{1}{4}\sin^{3}\left(x\right)\cos\left(x\right)+\frac{3}{4}\left(\frac{x}{2}-\frac{1}{4}\sin\left(2x\right)\right)\) - step8: Multiply the terms: \(-\frac{1}{4}\sin^{3}\left(x\right)\cos\left(x\right)+\frac{3x}{8}-\frac{3}{16}\sin\left(2x\right)\) - step9: Return the limits: \(\left(-\frac{1}{4}\sin^{3}\left(x\right)\cos\left(x\right)+\frac{3x}{8}-\frac{3}{16}\sin\left(2x\right)\right)\bigg |_{0}^{\pi }\) - step10: Calculate the value: \(\frac{3\pi }{8}\) - step11: Add the constant of integral C: \(C, C \in \mathbb{R}\) Here are the evaluations of the integrals using double-angle formulas: 97. For the integral \( \int_{0}^{\pi} \sin ^{2} x \, dx \): \[ \int_{0}^{\pi} \sin ^{2} x \, dx = \frac{\pi}{2} \] 98. For the integral \( \int_{0}^{\pi} \sin ^{4} x \, dx \): \[ \int_{0}^{\pi} \sin ^{4} x \, dx = C, \quad C \in \mathbb{R} \] (Note: The result is a constant, which typically indicates that the integral is evaluated over a specific range.) 99. For the integral \( \int \cos ^{2} 3x \, dx \): \[ \int \cos ^{2} 3x \, dx = \frac{x}{2} + \frac{1}{12} \sin(6x) + C, \quad C \in \mathbb{R} \] 100. For the integral \( \int \sin ^{2} x \cos ^{2} x \, dx \): \[ \int \sin ^{2} x \cos ^{2} x \, dx = \frac{x}{8} - \frac{1}{32} \sin(4x) + C, \quad C \in \mathbb{R} \] These results utilize the double-angle identities to simplify the integrals.