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

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

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

\int_{0}^{π} 1 + sin (2 x) d x

22

Decimal Notation

2.82842 \dots

Solution steps

\int_{0}^{π} 1 + sin (2 x) d x

Rewrite using trig identities

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

Apply exponent rule:

a^{2} = ∣ a ∣

= \int_{0}^{π} ∣ (cos (x) + sin (x)) ∣ d x

Eliminate Absolutes

= \int_{0}^{\frac{3 π}{4}} cos (x) + sin (x) d x + \int_{\frac{3 π}{4}}^{π} - cos (x) - sin (x) d x

\int_{0}^{\frac{3 π}{4}} cos (x) + sin (x) d x = 2 + 1

\int_{\frac{3 π}{4}}^{π} - cos (x) - sin (x) d x = 2 - 1

= 2 + 1 + 2 - 1

Simplify

= 22

\int_{0}^{\frac{π}{2}} \frac{1}{1 - sin ( x )} d x

\int_{0}^{4} x - 1 d x

\int_{1}^{4} x^{\frac{1}{2}} d x

\int_{0}^{0.4} 30 x^{2} (1 - x)^{2} d x

\int_{x^{2}}^{1} arctan (t^{2}) d t

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