Evaluate integral from 0 to pi of (x*sin(x)) / (1 + cos2(x)) dx and show that it equals pi2/4.
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Step-by-Step Solution
Step 1: Apply Property of Definite Integrals
We use the property of definite integrals that states ∫0af(x)dx=∫0af(a−x)dx. Here, a=π. We substitute x with (π−x) in the integrand. Note that sin(π−x)=sin(x) and cos(π−x)=−cos(x), so cos2(π−x)=(−cos(x))2=cos2(x).
Step 2: Simplify and Add the Integrals
Now we have two expressions for I. We add the original integral and the transformed integral. The denominators are the same, so we can combine the numerators. The x terms cancel out, leaving πsin(x) in the numerator.
Step 3: Substitute to Simplify the Integral
To evaluate the integral, we use a substitution. Let t=cos(x). Then, dt=−sin(x)dx. We also need to change the limits of integration. When x=0, t=cos(0)=1. When x=π, t=cos(π)=−1.
Step 4: Evaluate the Integral
We can change the order of integration by negating the integral: ∫abf(x)dx=−∫baf(x)dx. The integral of 1+t21 is tan−1(t). We evaluate this at the new limits, 1 and −1. We know that tan−1(1)=4π and tan−1(−1)=−4π.
Step 5: Solve for I
Finally, we divide both sides by 2 to find the value of $I.