What is the integral of xcos(xy) ?

The integral of xcos(xy) is sin(xy)+C

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integral of xcos(xy)

\int x cos (x y) d y

sin (x y) + C

Solution steps

\int 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 cos (x y) d y

Apply u-substitution

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

= x \cdot \int \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 cos (u) d u

Use the common integral:

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

= x \frac{1}{x} sin (u)

Substitute back

u = x y

= x \frac{1}{x} sin (x y)

Simplify

x \frac{1}{x} sin (x y) : sin (x y)

= sin (x y)

Add a constant to the solution

= sin (x y) + C

\frac{d}{d x} (ln (4 x^{2} + 1))

\frac{d}{d x} (1000 (1 - e^{- 0.15 x}))

3^{x} lo g_{3} (x)

\int \frac{1}{x - x ^{2}} d x

9 - x^{2}, - 8 x