What is the integral from 0 to pi/2 of 4cos(x/2) ?

The integral from 0 to pi/2 of 4cos(x/2) is 4sqrt(2)

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integral from 0 to pi/2 of 4cos(x/2)

\int_{0}^{\frac{π}{2}} 4 cos (\frac{x}{2}) d x

42

Decimal

5.65685 \dots

Solution steps

\int_{0}^{\frac{π}{2}} 4 cos (\frac{x}{2}) d x

Take the constant out:

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

= 4 \cdot \int_{0}^{\frac{π}{2}} cos (\frac{x}{2}) d x

Apply u-substitution

= 4 \cdot \int_{0}^{\frac{π}{4}} cos (u) \cdot 2 d u

Take the constant out:

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

= 4 \cdot 2 \cdot \int_{0}^{\frac{π}{4}} cos (u) d u

Use the common integral:

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

= 4 \cdot 2 [sin (u)]_{0}^{\frac{π}{4}}

Simplify

= 8 [sin (u)]_{0}^{\frac{π}{4}}

Compute the boundaries:

\frac{1}{2}

= 8 \cdot \frac{1}{2}

Simplify

= 42

What is the integral from 0 to pi/2 of 4cos(x/2) ?
The integral from 0 to pi/2 of 4cos(x/2) is 4sqrt(2)