4. [-/1 Points]
DETAILS SCALC9 4.5.012.
MY NOTES
ASK Y
Evaluate the indefinite integral. (Use $C$ for the constant of integration.)
\[
\int \sin (t) \sqrt{1+\cos (t)} d t
\]
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Answer
This is the simplest form of the answer, and it satisfies the requirements of the problem.
Step 1 ：First, we notice that the derivative of \(1 + \cos(t)\) is \(-\sin(t)\), which is present in the integral. This suggests a substitution.
Step 2 ：Let \(u = 1 + \cos(t)\). Then, \(du = -\sin(t) dt\).
Step 3 ：Substituting these into the integral, we get \(-\int \sqrt{u} du\).
Step 4 ：This is a standard power rule integral. The antiderivative of \(u^{n}\) is \(\frac{1}{n+1}u^{n+1}\).
Step 5 ：Applying this rule, we get \(-\frac{2}{3}u^{3/2}\).
Step 6 ：Substituting back for \(u\), we get \(-\frac{2}{3}(1 + \cos(t))^{3/2}\).
Step 7 ：Finally, we add the constant of integration, \(C\), to get the final answer: \(-\frac{2}{3}(1 + \cos(t))^{3/2} + C\).
Step 8 ：This is the simplest form of the answer, and it satisfies the requirements of the problem.
