What is the integral of xsin(xy) ?

The integral of xsin(xy) is -cos(xy)+C

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integral of xsin(xy)

\int x sin (x y) d y

- cos (x y) + C

Solution steps

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

Apply u-substitution

: \int \frac{sin ( u )}{x} d u

= x \cdot \int \frac{sin ( 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 sin (u) d u

Use the common integral:

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

= x \frac{1}{x} (- cos (u))

Substitute back

u = x y

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

Simplify

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

= - cos (x y)

Add a constant to the solution

= - cos (x y) + C

\frac{d}{d x} (cos^{3} (4 x))

x \to 2 lim (x^{3} + 2 x + 1)

(3, - 8), (- 5, - 7)

\int 7 x^{\frac{3}{2}} d x

y = \frac{3 x - 7}{2 x + 4}