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- Recognize the integral: Recognize the integral that needs to be solved.We need to find the integral of with respect to , which is written as .
- Use trigonometric identity: Use a trigonometric identity to simplify the integrand.We can use the power-reduction formula for sine, which states that . However, we need to apply this formula twice because we have .
- Apply power-reduction formula: Apply the power-reduction formula to . First, we apply it to : Now, we need to apply it again to get : $\sin^{\(4\)}(\(5\)x) = (\sin^{\(2\)}(\(5\)x))^\(2\) = \left(\frac{\(1\) - \cos(\(10\)x)}{\(2\)}\right)^\(2\)
- Expand binomial square: Expand the square of the binomial. \(\left(\frac{1 - \cos(10x)}{2}\right)^2 = \frac{1}{4}(1 - 2\cos(10x) + \cos^2(10x))\)
- Apply formula to \(\cos^2\): Apply the power-reduction formula again to \(\cos^2(10x)\).\(\newline\)\(\cos^2(10x) = \frac{1 + \cos(20x)}{2}\)\(\newline\)Substitute this back into the expanded binomial:\(\newline\)\(\frac{1}{4}(1 - 2\cos(10x) + \frac{1 + \cos(20x)}{2})\)
- Simplify before integrating: Simplify the expression before integrating.\(\newline\)\((\frac{1}{4})(1 - 2\cos(10x) + \frac{1 + \cos(20x)}{2}) = (\frac{1}{4})(1 - 2\cos(10x) + \frac{1}{2} + \frac{\cos(20x)}{2})\)\(\newline\)Combine like terms:\(\newline\)\(= (\frac{1}{4})(\frac{3}{2} - 2\cos(10x) + \frac{\cos(20x)}{2})\)\(\newline\)\(= (\frac{3}{8}) - (\frac{1}{2})\cos(10x) + (\frac{1}{8})\cos(20x)\)
- Integrate each term: Integrate each term separately.\(\newline\)\(\int((\frac{3}{8}) - (\frac{1}{2})\cos(10x) + (\frac{1}{8})\cos(20x))dx\)\(\newline\)= \((\frac{3}{8})x - (\frac{1}{2})\int\cos(10x)dx + (\frac{1}{8})\int\cos(20x)dx\)
- Integrate cosine terms: Integrate the cosine terms.\(\newline\)For \(\int \cos(10x)\,dx\), we use the substitution \(u = 10x\), \(du = 10dx\), so \(dx = \frac{du}{10}\).\(\newline\)\(\int \cos(10x)\,dx = \frac{1}{10}\int \cos(u)\,du = \frac{1}{10}\sin(u) + C = \frac{1}{10}\sin(10x) + C\)\(\newline\)For \(\int \cos(20x)\,dx\), we use the substitution \(u = 20x\), \(du = 20dx\), so \(dx = \frac{du}{20}\).\(\newline\)\(\int \cos(20x)\,dx = \frac{1}{20}\int \cos(u)\,du = \frac{1}{20}\sin(u) + C = \frac{1}{20}\sin(20x) + C\)\(\newline\)Now substitute these results back into the integral.
- Write final answer: Write the final answer.\(\newline\)The integral of \(\sin^{4}(5x)\) with respect to \(x\) is:\(\newline\)\[\int \sin^{4}(5x)\,dx = \left(\frac{3}{8}\right)x - \left(\frac{1}{2}\right)\left(\frac{1}{10}\right)\sin(10x) + \left(\frac{1}{8}\right)\left(\frac{1}{20}\right)\sin(20x) + C\]\(\newline\)Simplify the coefficients:\(\newline\)\[= \left(\frac{3}{8}\right)x - \left(\frac{1}{20}\right)\sin(10x) + \left(\frac{1}{160}\right)\sin(20x) + C\]
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