Evaluate the indefinite integral. (Use C for the constant of integration.)
\[
\int \sin (t) \sqrt{1+\cos (t)} d t
\]
Answer
Therefore, the solution to the indefinite integral \(\int \sin (t) \sqrt{1+\cos (t)} d t\) is \(\boxed{-\frac{2}{3} (1 + \cos(t))^{\frac{3}{2}} + C}\)
Step 1 ：Let's substitute \(u = 1 + \cos(t)\)
Step 2 ：Then, \(du = -\sin(t) dt\)
Step 3 ：Now, let's rewrite the integral using the substitution: \(\int \sin(t) \sqrt{1+\cos(t)} dt = \int -\sqrt{u} du\)
Step 4 ：To solve this new integral, we can use the power rule for integration: \(\int -\sqrt{u} du = -\frac{2}{3} u^{\frac{3}{2}} + C\)
Step 5 ：Substituting back \(u = 1 + \cos(t)\), we get: \(-\frac{2}{3} (1 + \cos(t))^{\frac{3}{2}} + C\)
Step 6 ：Therefore, the solution to the indefinite integral \(\int \sin (t) \sqrt{1+\cos (t)} d t\) is \(\boxed{-\frac{2}{3} (1 + \cos(t))^{\frac{3}{2}} + C}\)
