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

The integral from 1 to 2 of xcos(xy) is sin(2x)-sin(x)

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integral from 1 to 2 of xcos(xy)

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

sin (2 x) - sin (x)

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Simplify

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

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

Compute the boundaries:

sin (2 x) - sin (x)

= sin (2 x) - sin (x)

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