What is the integral from-1 to 1 of 2sqrt(1-(x^2)/2) ?

The integral from-1 to 1 of 2sqrt(1-(x^2)/2) is sqrt(2)(pi/2+1)

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integral from-1 to 1 of 2sqrt(1-(x^2)/2)

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

2 (\frac{π}{2} + 1)

Decimal

3.63565 \dots

Solution steps

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

Take the constant out:

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

= 2 \cdot \int_{- 1}^{1} 1 - \frac{x ^{2}}{2} d x

Apply Trigonometric Substitution

= 2 \cdot \int_{- \frac{π}{4}}^{\frac{π}{4}} 2 cos^{2} (u) d u

Take the constant out:

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

= 22 \cdot \int_{- \frac{π}{4}}^{\frac{π}{4}} cos^{2} (u) d u

Rewrite using trig identities

= 22 \cdot \int_{- \frac{π}{4}}^{\frac{π}{4}} \frac{1 + cos ( 2 u )}{2} d u

Take the constant out:

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

= 22 \frac{1}{2} \cdot \int_{- \frac{π}{4}}^{\frac{π}{4}} 1 + cos (2 u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 22 \frac{1}{2} (\int_{- \frac{π}{4}}^{\frac{π}{4}} 1 d u + \int_{- \frac{π}{4}}^{\frac{π}{4}} cos (2 u) d u)

\int_{- \frac{π}{4}}^{\frac{π}{4}} 1 d u = \frac{π}{2}

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

= 22 \frac{1}{2} (\frac{π}{2} + 1)

Simplify

22 \frac{1}{2} (\frac{π}{2} + 1) : 2 (\frac{π}{2} + 1)

= 2 (\frac{π}{2} + 1)

What is the integral from-1 to 1 of 2sqrt(1-(x^2)/2) ?
The integral from-1 to 1 of 2sqrt(1-(x^2)/2) is sqrt(2)(pi/2+1)