What is the integral of 2xycos(xy^2) ?

The integral of 2xycos(xy^2) is sin(xy^2)+C

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integral of 2xycos(xy^2)

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

sin (x y^{2}) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

Take the constant out:

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

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

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

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

Simplify

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

= sin (x y^{2})

Add a constant to the solution

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

What is the integral of 2xycos(xy^2) ?
The integral of 2xycos(xy^2) is sin(xy^2)+C