Evaluate the Integral

Problem

(∫_^)(sin(t)√(,1+cos(t))*d(t))

Solution

1. Identify a suitable substitution to simplify the integrand, choosing the expression under the radical.

u=1+cos(t)

2. Differentiate the substitution equation to find the relationship between d(u) and d(t)

d(u)/d(t)=−sin(t)

d(u)=−sin(t)*d(t)

−d(u)=sin(t)*d(t)

3. Substitute the expressions for u and d(u) into the original integral.

(∫_^)(√(,u)*(−d(u)))

−(∫_^)(u(1/2)*d(u))

4. Integrate using the power rule for integration.

−(u(3/2))/(3/2)+C

−2/3*u(3/2)+C

5. Back-substitute the original expression for u to get the final result in terms of t

−2/3*(1+cos(t))(3/2)+C

Final Answer

(∫_^)(sin(t)√(,1+cos(t))*d(t))=−2/3*(1+cos(t))(3/2)+C

Want more problems? Check here!