Encontrar la integral \( \int \cos ^{7} x d x \)

Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int \cos^{7}\left(x\right) dx\) - step1: Evaluate the integral: \(\frac{\cos^{7-1}\left(x\right)\sin\left(x\right)}{7}+\frac{7-1}{7}\times \int \cos^{7-2}\left(x\right) dx\) - step2: Subtract the numbers: \(\frac{\cos^{6}\left(x\right)\sin\left(x\right)}{7}+\frac{7-1}{7}\times \int \cos^{7-2}\left(x\right) dx\) - step3: Subtract the numbers: \(\frac{\cos^{6}\left(x\right)\sin\left(x\right)}{7}+\frac{7-1}{7}\times \int \cos^{5}\left(x\right) dx\) - step4: Subtract the numbers: \(\frac{\cos^{6}\left(x\right)\sin\left(x\right)}{7}+\frac{6}{7}\times \int \cos^{5}\left(x\right) dx\) - step5: Evaluate the integral: \(\frac{\cos^{6}\left(x\right)\sin\left(x\right)}{7}+\frac{6}{7}\left(\frac{1}{5}\cos^{4}\left(x\right)\sin\left(x\right)+\frac{4}{15}\cos^{2}\left(x\right)\sin\left(x\right)+\frac{8}{15}\sin\left(x\right)\right)\) - step6: Simplify the expression: \(\frac{1}{7}\cos^{6}\left(x\right)\sin\left(x\right)+\frac{6}{7}\left(\frac{1}{5}\cos^{4}\left(x\right)\sin\left(x\right)+\frac{4}{15}\cos^{2}\left(x\right)\sin\left(x\right)+\frac{8}{15}\sin\left(x\right)\right)\) - step7: Multiply the terms: \(\frac{1}{7}\cos^{6}\left(x\right)\sin\left(x\right)+\frac{6}{35}\cos^{4}\left(x\right)\sin\left(x\right)+\frac{8}{35}\cos^{2}\left(x\right)\sin\left(x\right)+\frac{16}{35}\sin\left(x\right)\) - step8: Add the constant of integral C: \(\frac{1}{7}\cos^{6}\left(x\right)\sin\left(x\right)+\frac{6}{35}\cos^{4}\left(x\right)\sin\left(x\right)+\frac{8}{35}\cos^{2}\left(x\right)\sin\left(x\right)+\frac{16}{35}\sin\left(x\right) + C, C \in \mathbb{R}\) La integral indefinida de \( \cos^{7} x \) es: \[ \int \cos^{7} x \, dx = \frac{1}{7}\cos^{6}(x)\sin(x) + \frac{6}{35}\cos^{4}(x)\sin(x) + \frac{8}{35}\cos^{2}(x)\sin(x) + \frac{16}{35}\sin(x) + C, \quad C \in \mathbb{R} \] Donde \( C \) es la constante de integración.