What is the integral from 0 to 1 of 2cos((pit)/2) ?

The integral from 0 to 1 of 2cos((pit)/2) is 4/pi

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integral from 0 to 1 of 2cos((pit)/2)

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

\frac{4}{π}

Decimal

1.27323 \dots

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int_{0}^{\frac{π}{2}} cos (u) \frac{2}{π} d u

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

Take the constant out:

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

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

Use the common integral:

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

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

Simplify

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

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

Compute the boundaries:

1

= \frac{4}{π} \cdot 1

Simplify

= \frac{4}{π}

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What is the integral from 0 to 1 of 2cos((pit)/2) ?
The integral from 0 to 1 of 2cos((pit)/2) is 4/pi