What is the integral from 0 to pi of sqrt(1+cos(2x)) ?

The integral from 0 to pi of sqrt(1+cos(2x)) is 2sqrt(2)

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integral from 0 to pi of sqrt(1+cos(2x))

\int_{0}^{π} 1 + cos (2 x) d x

22

Decimal

2.82842 \dots

Solution steps

\int_{0}^{π} 1 + cos (2 x) d x

Rewrite using trig identities

= \int_{0}^{π} 2 cos^{2} (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{0}^{π} cos^{2} (x) d x

Apply exponent rule:

a^{2} = ∣ a ∣

= 2 \cdot \int_{0}^{π} ∣ cos (x) ∣ d x

Eliminate Absolutes

= 2 (\int_{0}^{\frac{π}{2}} cos (x) d x + \int_{\frac{π}{2}}^{π} - cos (x) d x)

\int_{0}^{\frac{π}{2}} cos (x) d x = 1

\int_{\frac{π}{2}}^{π} - cos (x) d x = 1

= 2 (1 + 1)

Simplify

= 22

What is the integral from 0 to pi of sqrt(1+cos(2x)) ?
The integral from 0 to pi of sqrt(1+cos(2x)) is 2sqrt(2)