Integral Sine Cosine 73F830

1. **State the problem:** We want to evaluate the integral $$\int \frac{\sin(x)}{\sqrt{\cos^2(x)+1}}\, dx.$$\n\n2. **Identify a substitution:** Notice the denominator involves $$\cos^2(x)+1$$ and the numerator is $$\sin(x)$$, which is the derivative of $$-\cos(x)$$. This suggests using $$u = \cos(x)$$.\n\n3. **Substitute variables:** Then $$du = -\sin(x) dx$$ or equivalently $$-du = \sin(x) dx$$.\n\n4. **Rewrite the integral:** Substitute into the integral: $$\int \frac{\sin(x)}{\sqrt{\cos^2(x)+1}} dx = \int \frac{-du}{\sqrt{u^2 + 1}} = -\int \frac{du}{\sqrt{u^2 + 1}}.$$\n\n5. **Recall the integral formula:** The integral $$\int \frac{du}{\sqrt{u^2 + a^2}} = \ln|u + \sqrt{u^2 + a^2}| + C$$ for $$a > 0$$. Here, $$a=1$$.\n\n6. **Evaluate the integral:** So, $$-\int \frac{du}{\sqrt{u^2 + 1}} = -\ln|u + \sqrt{u^2 + 1}| + C.$$\n\n7. **Back-substitute:** Replace $$u = \cos(x)$$ back: $$-\ln|\cos(x) + \sqrt{\cos^2(x) + 1}| + C.$$\n\n**Final answer:** $$\boxed{-\ln|\cos(x) + \sqrt{\cos^2(x) + 1}| + C}.$$