Evaluate the indefinite integral. (use C for the constant of integration. ∫ sin(t)√(1 + cos(t)) dt

Solution:

Given, ∫ sin(t) √(1 + cos(t)) dt

Let 1 + cos(t) dt = u

Then, du = -sin(t) dt

So, dt = (-1/sin(t))du

Now, the given integral becomes

= \(\int -1\sqrt{u}du\)

= -1\(\int \sqrt{u}du\)

= \(-\int u^{\frac{1}{2}}du\)

= \(-\frac{2}{3}u^{\frac{3}{2}}+C\)

Now replacing u with 1 + cos(t)

= \(-\frac{2}{3}(1+cos(t))^{\frac{3}{2}}+C\)

Therefore, the solution of the indefinite integral is \(-\frac{2}{3}(1+cos(t))^{\frac{3}{2}}+C\).

Evaluate the indefinite integral. (use C for the constant of integration. ∫ sin(t) √(1 + cos(t)) dt

Summary:

The general solution of the indefinite integral ∫ sin(t) √(1 + cos(t)) dt is \(-\frac{2}{3}(1+cos(t))^{\frac{3}{2}}+C\).