What is the integral of sin(3x^2-7)x ?

The integral of sin(3x^2-7)x is -1/6 cos(3x^2-7)+C

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integral of sin(3x^2-7)x

\int sin (3 x^{2} - 7) x d x

- \frac{1}{6} cos (3 x^{2} - 7) + C

Solution steps

\int sin (3 x^{2} - 7) x d x

Apply u-substitution

: \int \frac{sin ( 3 u - 7 )}{2} d u

= \int \frac{sin ( 3 u - 7 )}{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 (3 u - 7) d u

Apply u-substitution

: \int sin (v) \frac{1}{3} d v

= \frac{1}{2} \cdot \int sin (v) \frac{1}{3} d v

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{3} \cdot \int sin (v) d v

Use the common integral:

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

= \frac{1}{2} \cdot \frac{1}{3} (- cos (v))

Substitute back

= \frac{1}{2} \cdot \frac{1}{3} (- cos (3 x^{2} - 7))

Simplify

\frac{1}{2} \cdot \frac{1}{3} (- cos (3 x^{2} - 7)) : - \frac{1}{6} cos (3 x^{2} - 7)

= - \frac{1}{6} cos (3 x^{2} - 7)

Add a constant to the solution

= - \frac{1}{6} cos (3 x^{2} - 7) + C

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What is the integral of sin(3x^2-7)x ?
The integral of sin(3x^2-7)x is -1/6 cos(3x^2-7)+C