What is the integral from 1/2 to 1 of sin(2pix) ?

The integral from 1/2 to 1 of sin(2pix) is -1/pi

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integral from 1/2 to 1 of sin(2pix)

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

- \frac{1}{π}

Decimal Notation

- 0.31830 \dots

Solution steps

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

Apply u-substitution

: \int_{π}^{2 π} sin (u) \frac{1}{2 π} d u

= \int_{π}^{2 π} sin (u) \frac{1}{2 π} d u

Take the constant out:

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

= \frac{1}{2 π} \cdot \int_{π}^{2 π} sin (u) d u

Use the common integral:

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

= \frac{1}{2 π} [- cos (u)]_{π}^{2 π}

Compute the boundaries:

- 2

= \frac{1}{2 π} (- 2)

Simplify

= - \frac{1}{π}

y = - x^{2} + 4, y = 2 x + 1

500000 + 5000 t^{2}

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What is the integral from 1/2 to 1 of sin(2pix) ?
The integral from 1/2 to 1 of sin(2pix) is -1/pi