What is the integral of xsin(9x^2) ?

The integral of xsin(9x^2) is -1/18 cos(9x^2)+C

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integral of xsin(9x^2)

\int x sin (9 x^{2}) d x

- \frac{1}{18} cos (9 x^{2}) + C

Solution steps

\int x sin (9 x^{2}) d x

Apply u-substitution

: \int \frac{sin ( u )}{18} d u

= \int \frac{sin ( u )}{18} d u

Take the constant out:

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

= \frac{1}{18} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{1}{18} (- cos (u))

Substitute back

u = 9 x^{2}

= \frac{1}{18} (- cos (9 x^{2}))

Simplify

= - \frac{1}{18} cos (9 x^{2})

Add a constant to the solution

= - \frac{1}{18} cos (9 x^{2}) + C

\frac{d}{d x} (\frac{x ^{6}}{6} + \frac{1}{16 x ^{4}})

cos (10 x) - 1

x \to 0 lim (\frac{sin ( 5 x )}{sin ( 9 x )})

\int x (3 x^{2} - 5) d x

3 x^{2} - 2 y^{3} \frac{d y}{d x} = 0

What is the integral of xsin(9x^2) ?
The integral of xsin(9x^2) is -1/18 cos(9x^2)+C