4. Evaluate the indefinite integral. (Use \( C \) for the constant of integration.) \[ \int \frac{\cos \left(\frac{\pi}{x^{31}}\right)}{x^{32}} d x \]

Calculate the integral \( \int \frac{\cos \left(\frac{\pi}{x^{31}}\right)}{x^{32}} d x \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int \frac{\cos\left(\frac{\pi }{x^{31}}\right)}{x^{32}} dx\) - step1: Use the substitution \(dx=-\frac{x^{32}}{31\pi } dt\) to transform the integral\(:\) \(\int \frac{\cos\left(\frac{\pi }{x^{31}}\right)}{x^{32}}\times \left(-\frac{x^{32}}{31\pi }\right) dt\) - step2: Simplify: \(\int -\frac{\cos\left(\frac{\pi }{x^{31}}\right)}{31\pi } dt\) - step3: Use the substitution \(t=\frac{\pi }{x^{31}}\) to transform the integral\(:\) \(\int -\frac{\cos\left(t\right)}{31\pi } dt\) - step4: Simplify the expression: \(\int -\frac{1}{31\pi }\times \cos\left(t\right) dt\) - step5: Use properties of integrals: \(-\frac{1}{31\pi }\times \int \cos\left(t\right) dt\) - step6: Evaluate the integral: \(-\frac{1}{31\pi }\times \sin\left(t\right)\) - step7: Substitute back: \(-\frac{1}{31\pi }\times \sin\left(\frac{\pi }{x^{31}}\right)\) - step8: Rewrite the expression: \(-\frac{\sin\left(\frac{\pi }{x^{31}}\right)}{31\pi }\) - step9: Add the constant of integral C: \(-\frac{\sin\left(\frac{\pi }{x^{31}}\right)}{31\pi } + C, C \in \mathbb{R}\) The indefinite integral of \( \frac{\cos \left(\frac{\pi}{x^{31}}\right)}{x^{32}} \) with respect to \( x \) is \( -\frac{\sin\left(\frac{\pi}{x^{31}}\right)}{31\pi} + C \), where \( C \) is the constant of integration.