What is the integral of y*sin(y^2) ?

The integral of y*sin(y^2) is -1/2 cos(y^2)+C

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integral of y*sin(y^2)

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

- \frac{1}{2} cos (y^{2}) + C

Solution steps

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

Apply u-substitution

: \int \frac{sin ( u )}{2} d u

= \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

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

Use the common integral:

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

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

Substitute back

u = y^{2}

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

Simplify

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

Add a constant to the solution

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

y = x^{3}, y = x

n = 0 \sum \infty n (\frac{1}{2})^{n}

\frac{d}{d x} (\frac{1}{y} e^{x y} - ln (y^{2}))

\frac{d}{d x} (7 x^{\frac{1}{2}} + x^{\frac{3}{2}})

n = 1 \sum \infty arctan (2 n)

What is the integral of y*sin(y^2) ?
The integral of y*sin(y^2) is -1/2 cos(y^2)+C