7. Evaluate the indefinite integral. (Use \( C \) for the constant of integration.) \[ \int \sin (t) \sqrt{1+\cos (t)} d t \]

Calculate the integral \( \int \sin(t) \sqrt{1+\cos(t)} dt \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int \sin\left(t\right)\times \sqrt{1+\cos\left(t\right)} dt\) - step1: Rewrite the expression: \(\int \sin\left(t\right)\left(1+\cos\left(t\right)\right)^{\frac{1}{2}} dt\) - step2: Use the substitution \(dt=-\frac{1}{\sin\left(t\right)} dv\) to transform the integral\(:\) \(\int \sin\left(t\right)\left(1+\cos\left(t\right)\right)^{\frac{1}{2}}\left(-\frac{1}{\sin\left(t\right)}\right) dv\) - step3: Simplify: \(\int -\left(1+\cos\left(t\right)\right)^{\frac{1}{2}} dv\) - step4: Use the substitution \(v=\cos\left(t\right)\) to transform the integral\(:\) \(\int -\left(1+v\right)^{\frac{1}{2}} dv\) - step5: Use properties of integrals: \(-\int \left(1+v\right)^{\frac{1}{2}} dv\) - step6: Use the substitution \(dv=1 dt\) to transform the integral\(:\) \(-\int \left(1+v\right)^{\frac{1}{2}}\times 1 dt\) - step7: Simplify: \(-\int \left(1+v\right)^{\frac{1}{2}} dt\) - step8: Use the substitution \(t=1+v\) to transform the integral\(:\) \(-\int t^{\frac{1}{2}} dt\) - step9: Evaluate the integral: \(-\frac{t^{\frac{1}{2}+1}}{\frac{1}{2}+1}\) - step10: Simplify: \(-\frac{t^{\frac{3}{2}}}{\frac{3}{2}}\) - step11: Substitute back: \(-\frac{\left(1+v\right)^{\frac{3}{2}}}{\frac{3}{2}}\) - step12: Substitute back: \(-\frac{\left(1+\cos\left(t\right)\right)^{\frac{3}{2}}}{\frac{3}{2}}\) - step13: Rewrite the expression: \(2\left(1+\cos\left(t\right)\right)^{\frac{3}{2}}\left(-\frac{1}{3}\right)\) - step14: Rewrite the expression: \(-\frac{2}{3}\left(1+\cos\left(t\right)\right)^{\frac{3}{2}}\) - step15: Simplify: \(-\frac{2}{3}\sqrt{\left(1+\cos\left(t\right)\right)^{3}}\) - step16: Add the constant of integral C: \(-\frac{2}{3}\sqrt{\left(1+\cos\left(t\right)\right)^{3}} + C, C \in \mathbb{R}\) The indefinite integral of \( \sin(t) \sqrt{1+\cos(t)} \) with respect to \( t \) is \( -\frac{2}{3}\sqrt{(1+\cos(t))^3} + C \), where \( C \) is the constant of integration.