Evaluate the integral: \int y\sin(xy)\;dx

Question:

Evaluate the integral:

{eq}\displaystyle \int y\sin(xy)\;dx {/eq}

Integration by substitution:

An integral without boundaries is called indefinite integral. To find the solution of the integral applies the integration by substitution method. It is also called a u-substitution method. Consider, {eq}u = xy {/eq} while applying integration by substitution.

{eq}\text{Solution}: \\ \text{Given}: \\ \displaystyle \int y\sin(xy)\;dx \\ \text{Apply integration by substitution method}: \\ u = yx \\ du = y\;dx \\ \text{Therefore}, \\ \begin{align*} \int y\sin(xy)\;dx &= y \int \frac{\sin(u)}{y}\;du \\ &= y\frac{1}{y} \int \sin(u)\;du & \left ( \text{Take the constant out} \right ) \\ &= \int \sin(u)\;du \\ &= - \cos(u) & \left ( \text{Use the common integral} \right ) \\ &= - \cos(yx) & \left ( \text{Where}, \; u = yx \right ) \\ &= - \cos(yx) + C & \left ( \text{Add the constant term to the solution} \right ) \\ \end{align*} \\ \boxed{\text{The solution of the integral} \; \displaystyle {\color{Blue}{\int y\sin(xy)\;dx = - \cos(yx) + C }}} {/eq}