What is the integral of 2xy*cos(x^2y) ?

The integral of 2xy*cos(x^2y) is sin(yx^2)+C

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integral of 2xy*cos(x^2y)

\int 2 x y \cdot cos (x^{2} y) d x

sin (y x^{2}) + C

Solution steps

\int 2 x y cos (x^{2} y) d x

Take the constant out:

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

= 2 y \cdot \int cos (y x^{2}) x d x

Apply u-substitution

: \int \frac{cos ( y u )}{2} d u

= 2 y \cdot \int \frac{cos ( y u )}{2} d u

Take the constant out:

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

= 2 y \frac{1}{2} \cdot \int cos (y u) d u

Apply u-substitution

: \int \frac{cos ( v )}{y} d v

= 2 y \frac{1}{2} \cdot \int \frac{cos ( v )}{y} d v

Take the constant out:

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

= 2 y \frac{1}{2} \cdot \frac{1}{y} \cdot \int cos (v) d v

Use the common integral:

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

= 2 y \frac{1}{2} \cdot \frac{1}{y} sin (v)

Substitute back

= 2 y \frac{1}{2} \cdot \frac{1}{y} sin (y x^{2})

Simplify

2 y \frac{1}{2} \cdot \frac{1}{y} sin (y x^{2}) : sin (y x^{2})

= sin (y x^{2})

Add a constant to the solution

= sin (y x^{2}) + C

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What is the integral of 2xy*cos(x^2y) ?
The integral of 2xy*cos(x^2y) is sin(yx^2)+C