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

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

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

\int_{0}^{1} 1 + cos (x) d x

22 sin (\frac{1}{2})

Decimal

1.35602 \dots

Solution steps

\int_{0}^{1} 1 + cos (x) d x

Apply u-substitution

= \int_{0}^{t a n (\frac{1}{2})} \frac{2 2}{( 1 + u ^{2} ) 1 + u ^{2}} d u

Take the constant out:

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

= 22 \cdot \int_{0}^{t a n (\frac{1}{2})} \frac{1}{( 1 + u ^{2} ) 1 + u ^{2}} d u

Simplify

(1 + u^{2}) 1 + u^{2} : (1 + u^{2})^{\frac{3}{2}}

= 22 \cdot \int_{0}^{t a n (\frac{1}{2})} \frac{1}{( 1 + u ^{2} ) ^{\frac{3}{2}}} d u

Apply Trigonometric Substitution

= 22 \cdot \int_{0}^{\frac{1}{2}} cos (v) d v

Use the common integral:

\int cos (v) d v = sin (v)

= 22 [sin (v)]_{0}^{\frac{1}{2}}

Compute the boundaries:

sin (\frac{1}{2})

= 22 sin (\frac{1}{2})

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