What is the integral of sin(xy)+xycos(xy) ?

The integral of sin(xy)+xycos(xy) is xsin(yx)+C

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integral of sin(xy)+xycos(xy)

\int sin (x y) + x y cos (x y) d x

x sin (y x) + C

Solution steps

\int sin (x y) + x y cos (x y) d x

Apply u-substitution

: \int \frac{sin ( u ) + u cos ( u )}{y} d u

= \int \frac{sin ( u ) + u cos ( u )}{y} d u

Take the constant out:

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

= \frac{1}{y} \cdot \int sin (u) + u cos (u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{y} (\int sin (u) d u + \int u cos (u) d u)

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

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

= \frac{1}{y} (- cos (u) + u sin (u) + cos (u))

Substitute back

u = y x

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

Simplify

\frac{1}{y} (- cos (y x) + y x sin (y x) + cos (y x)) : x sin (y x)

= x sin (y x)

Add a constant to the solution

= x sin (y x) + C

\int_{3}^{7} \frac{( 2 x - 3 )}{6} d x

\int \frac{x + 6}{5 - 4 x - x ^{2}} d x

\frac{d}{d t} (ln (cos (\frac{π}{2} - \frac{t}{2})))

\int (x + 2) e^{4 - 3 x} d x

n = 1 \sum \infty \frac{3 ^{n}}{n}

What is the integral of sin(xy)+xycos(xy) ?
The integral of sin(xy)+xycos(xy) is xsin(yx)+C