Evaluate the definite integral ∫_0^{π} cos^2 x dx.

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Multiple Choice

Evaluate the definite integral ∫_0^{π} cos^2 x dx.

The idea is to simplify cos^2 x using a standard identity and then integrate term by term. Use cos^2 x = (1 + cos 2x)/2. So ∫ from 0 to π of cos^2 x dx = ∫0^π (1 + cos 2x)/2 dx = 1/2 ∫0^π dx + 1/2 ∫0^π cos 2x dx = 1/2 [x]0^π + 1/2 [ (1/2) sin 2x ]0^π = π/2 + 0 = π/2. Thus the value is π/2. The cos^2 x function is always nonnegative and has average value 1/2 over a full period, so over an interval of length π it integrates to π/2.

The idea is to simplify cos^2 x using a standard identity and then integrate term by term. Use cos^2 x = (1 + cos 2x)/2. So

∫ from 0 to π of cos^2 x dx = ∫0^π (1 + cos 2x)/2 dx

= 1/2 ∫0^π dx + 1/2 ∫0^π cos 2x dx

= 1/2 [x]0^π + 1/2 [ (1/2) sin 2x ]0^π

= π/2 + 0 = π/2.

Thus the value is π/2. The cos^2 x function is always nonnegative and has average value 1/2 over a full period, so over an interval of length π it integrates to π/2.

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