What is the integral of 2ysin(y^2) ?

The integral of 2ysin(y^2) is -cos(y^2)+C

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integral of 2ysin(y^2)

\int 2 y sin (y^{2}) d y

- cos (y^{2}) + C

Solution steps

\int 2 y sin (y^{2}) d y

Take the constant out:

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

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

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

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

= 2 \cdot \frac{1}{2} (- cos (u))

Substitute back

u = y^{2}

= 2 \cdot \frac{1}{2} (- cos (y^{2}))

Simplify

2 \cdot \frac{1}{2} (- cos (y^{2})) : - cos (y^{2})

= - cos (y^{2})

Add a constant to the solution

= - cos (y^{2}) + C