**Step 1: Identify the Substitution and Its Differential** Substitution given: \( u = \cos x \) Differential: \( du = -\sin x \, dx \) Or, \( -du = \sin x \, dx \) **Explanation:** Substitution allows transformation of the variable \(x\) into \(u\), which simplifies integrals involving \(\cos x\) and \(\sin x\) in a certain way. The derivative \( du = -\sin x \, dx \) indicates that whenever \(\sin x\) appears in the integrand along with \(dx\), it can be replaced by \(-du\). --- **Step 2: Analyze Each Integral for Compatibility with the Substitution** Look for integrals where: - \(\sin x \, dx\) is present (so it can be replaced by \(-du\)), - the rest of the integrand can be written in terms of \(\cos x\) (so it becomes a function of \(u\)). Go through each option: 1. \( \int \frac{\cos x}{1 + \sin^2 x} dx \): Substitution does not directly fit; denominator involves \(\sin^2 x\). 2. \( \int (\sin^{2025} x) \cos x \, dx \): \(\cos x \, dx\) does not match \(du\). 3. \( \int e^{\sin x} dx \): No matching \(\sin x \, dx\). 4. \( \int \frac{\sin x}{\cos^{2025} x} dx \): \(\sin x \, dx\) can be replaced by \(-du\), and \(\cos^{2025} x = u^{2025}\). The integral becomes \( \int \frac{1}{u^{2025}} (-du) \), which is fully in terms of \(u\). 5. \( \int e^{\cos x} \sin x \, dx \): \(\sin x \, dx = -du\), and \(e^{\cos x} = e^u\). The integral becomes \( \int e^u (-du) \), which is also in terms of \(u\). 6. \( \int \frac{\cos x}{\sin^2 x} dx \): \(\sin^2 x\) in denominator, but differential does not match. 7. \( \int e^{\sin x} \cos x \, dx \): \(\cos x \, dx\) does not match \(du\). 8. \( \int (\cos^{2025} x) \sin x \, dx \): \(\sin x \, dx = -du\), and \(\cos^{2025} x = u^{2025}\). The integral becomes \( \int u^{2025} (-du) \), in terms of \(u\). 9. \( \int \frac{\sin x}{1 + \cos^2 x} dx \): \(\sin x \, dx = -du\), and \(1 + \cos^2 x = 1 + u^2\). The integral becomes \( \int \frac{1}{1+u^2} (-du) \), in terms of \(u\). **Explanation:** Any integral where \(\sin x \, dx\) appears and the rest of the function is a function of \(\cos x\) can be directly converted into an integral in terms of \(u\). --- **Final Answer Summary:** The substitution \(u = \cos x,\ du = -\sin x\, dx\) helps to integrate the following options: - \( \int \frac{\sin x}{\cos^{2025} x} dx \) - \( \int e^{\cos x} \sin x \, dx \) - \( \int (\cos^{2025} x) \sin x \, dx \) - \( \int \frac{\sin x}{1 + \cos^2 x} dx \) **Definition of Parameters:** - **Relevancy:** Options directly using the substitution \(u = \cos x\) and \(du = -\sin x dx\). - **Completeness:** All valid options are listed, with explanation for each. - **Accuracy:** Calculations and reasoning are shown step by step. - **Clarity:** Each step and option is explained clearly. - **Structure:** The answer has organized steps and a final summary. - **Voice:** The language is clear, direct, and human-like.