Question

Integral with limits 0 to π/2 of sinx/√1+cosx

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A) 1
B) 0.5
C) 1/√2
D) √2/2

Answer 1

To solve the integral, we can use the substitution method. After evaluating the integral, the answer is approximately 0.8284, or √2/2.

Step-by-step explanation:

To solve this integral, we can use the substitution method. Let's substitute u = 1 + cos(x). Differentiating both sides, we get du = -sin(x) dx. Notice that the numerator of the integrand, sin(x), is equal to -du. The denominator remains as √(1 + cos(x)). Now we can rewrite the integral as: ∫(-1/√u) du. Integrate this expression using the power rule for integration, which gives us -2√u. Substituting back u = 1 + cos(x) into the result, we obtain -2√(1 + cos(x)).

Now, let's evaluate the integral from 0 to π/2:

1. 
   
3. 
   
5. 
   

Simplifying this expression gives us -2 + 2√2, which is approximately 0.8284. Therefore, the correct option is D) √2/2.