What is the integral of sin(2x^2)2x ?

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

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integral of sin(2x^2)2x

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{sin ( u )}{4} d u

= 2 \cdot \int \frac{sin ( u )}{4} d u

Take the constant out:

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

= 2 \cdot \frac{1}{4} \cdot \int sin (u) d u

Use the common integral:

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

= 2 \cdot \frac{1}{4} (- cos (u))

Substitute back

u = 2 x^{2}

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

Simplify

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

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

Add a constant to the solution

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

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