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

The integral from 0 to pi of ysin(xy) is (sin(pix)-pixcos(pix))/(x^2)

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integral from 0 to pi of ysin(xy)

\int_{0}^{π} y sin (x y) d y

\frac{sin ( π x ) - π x cos ( π x )}{x ^{2}}

Solution steps

\int_{0}^{π} y sin (x y) d y

Apply Integration By Parts

: [\frac{1}{x} (- y cos (x y) - x \cdot \int - \frac{1}{x} cos (x y) d y)]_{0}^{π}

= [\frac{1}{x} (- y cos (x y) - x \cdot \int - \frac{1}{x} cos (x y) d y)]_{0}^{π}

\int - \frac{1}{x} cos (x y) d y = - \frac{1}{x ^{2}} sin (x y)

= [\frac{1}{x} (- y cos (x y) - x (- \frac{1}{x ^{2}} sin (x y)))]_{0}^{π}

Simplify

[\frac{1}{x} (- y cos (x y) - x (- \frac{1}{x ^{2}} sin (x y)))]_{0}^{π} : [\frac{1}{x} (- y cos (x y) + \frac{1}{x} sin (x y))]_{0}^{π}

= [\frac{1}{x} (- y cos (x y) + \frac{1}{x} sin (x y))]_{0}^{π}

Compute the boundaries:

\frac{1}{x} (\frac{1}{x} sin (π x) - π cos (π x))

= \frac{1}{x} (\frac{1}{x} sin (π x) - π cos (π x))

Simplify

= \frac{sin ( π x ) - π x cos ( π x )}{x ^{2}}

\int_{0}^{1} x^{4} - x^{2} d x

\int_{0}^{\frac{π}{4}} \frac{d}{1} (cos^{4} (x)) d x

\int_{2}^{6} (x^{2} + 7 x - 5) (x - 2) d x

\int_{1}^{2} (x - 1) (2 - x) d x

\int_{- \frac{π}{4}}^{\frac{π}{4}} sin^{2} (x) d x

What is the integral from 0 to pi of ysin(xy) ?
The integral from 0 to pi of ysin(xy) is (sin(pix)-pixcos(pix))/(x^2)