What is the integral of 4xsin(4x^2) ?

The integral of 4xsin(4x^2) is -1/2 cos(4x^2)+C

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integral of 4xsin(4x^2)

\int 4 x sin (4 x^{2}) d x

- \frac{1}{2} cos (4 x^{2}) + C

Solution steps

\int 4 x sin (4 x^{2}) d x

Take the constant out:

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

= 4 \cdot \int x sin (4 x^{2}) d x

Apply u-substitution

: \int \frac{sin ( u )}{8} d u

= 4 \cdot \int \frac{sin ( u )}{8} d u

Take the constant out:

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

= 4 \cdot \frac{1}{8} \cdot \int sin (u) d u

Use the common integral:

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

= 4 \cdot \frac{1}{8} (- cos (u))

Substitute back

u = 4 x^{2}

= 4 \cdot \frac{1}{8} (- cos (4 x^{2}))

Simplify

4 \cdot \frac{1}{8} (- cos (4 x^{2})) : - \frac{1}{2} cos (4 x^{2})

= - \frac{1}{2} cos (4 x^{2})

Add a constant to the solution

= - \frac{1}{2} cos (4 x^{2}) + C

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What is the integral of 4xsin(4x^2) ?
The integral of 4xsin(4x^2) is -1/2 cos(4x^2)+C