Evaluate the integral from 0 to pi/2 of sin5(x)/(sin5(x)+cos5(x))dx.
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Step-by-Step Solution
Step 1: Define the integral
Let the given integral be denoted by I. This is the standard way to begin solving definite integral problems.
Step 2: Apply the property of definite integrals
We use the property of definite integrals which states that ∫0af(x)dx=∫0af(a−x)dx. Here, a=2π. Applying this property, sin(2π−x)=cos(x) and cos(2π−x)=sin(x).
Step 3: Simplify the integral
After applying the property, the integral transforms into this new form. We now have two expressions for $I.
Step 4: Add the two integral expressions
Adding the original integral (from Step 1) and the transformed integral (from Step 3) gives 2I. The denominators are the same, so we can combine the numerators.
Step 5: Simplify and evaluate the integral
The numerator and denominator cancel out, leaving an integral of 1 with respect to x. Integrating 1 gives x. Evaluating x from 0 to 2π yields 2π.
Step 6: Solve for I
Since 2I=2π, we divide by 2 to find the value of $I.