What is the integral from 0 to 1 of ysin(xy) ?

The integral from 0 to 1 of ysin(xy) is -cos(y)+1

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integral from 0 to 1 of ysin(xy)

\int_{0}^{1} y sin (x y) d x

- cos (y) + 1

Solution steps

\int_{0}^{1} y sin (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_{0}^{1} sin (y x) d x

Apply u-substitution

= y \cdot \int_{0}^{y} \frac{sin ( 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_{0}^{y} sin (u) d u

Use the common integral:

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

= y \frac{1}{y} [- cos (u)]_{0}^{y}

Simplify

y \frac{1}{y} [- cos (u)]_{0}^{y} : [- cos (u)]_{0}^{y}

= [- cos (u)]_{0}^{y}

Compute the boundaries:

- cos (y) + 1

= - cos (y) + 1

What is the integral from 0 to 1 of ysin(xy) ?
The integral from 0 to 1 of ysin(xy) is -cos(y)+1