What is the integral from 1 to 2 of ycos(xy) ?

The integral from 1 to 2 of ycos(xy) is sin(2y)-sin(y)

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integral from 1 to 2 of ycos(xy)

\int_{1}^{2} y cos (x y) d x

sin (2 y) - sin (y)

Solution steps

\int_{1}^{2} y cos (x y) d x

Take the constant out:

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

= y \cdot \int_{1}^{2} cos (y x) d x

Apply u-substitution

: \int_{y}^{y \cdot 2} \frac{cos ( u )}{y} d u

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

Take the constant out:

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

= y \frac{1}{y} \cdot \int_{y}^{y \cdot 2} cos (u) d u

Use the common integral:

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

= y \frac{1}{y} [sin (u)]_{y}^{y \cdot 2}

Simplify

y \frac{1}{y} [sin (u)]_{y}^{y \cdot 2} : [sin (u)]_{y}^{2 y}

= [sin (u)]_{y}^{2 y}

Compute the boundaries:

sin (2 y) - sin (y)

= sin (2 y) - sin (y)

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What is the integral from 1 to 2 of ycos(xy) ?
The integral from 1 to 2 of ycos(xy) is sin(2y)-sin(y)