What is the integral of 20xsin(7x^2-8) ?

The integral of 20xsin(7x^2-8) is -10/7 cos(7x^2-8)+C

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integral of 20xsin(7x^2-8)

\int 20 x sin (7 x^{2} - 8) d x

- \frac{10}{7} cos (7 x^{2} - 8) + C

Solution steps

\int 20 x sin (7 x^{2} - 8) d x

Take the constant out:

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

= 20 \cdot \int x sin (7 x^{2} - 8) d x

Apply u-substitution

: \int \frac{sin ( 7 u - 8 )}{2} d u

= 20 \cdot \int \frac{sin ( 7 u - 8 )}{2} d u

Take the constant out:

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

= 20 \cdot \frac{1}{2} \cdot \int sin (7 u - 8) d u

Apply u-substitution

: \int sin (v) \frac{1}{7} d v

= 20 \cdot \frac{1}{2} \cdot \int sin (v) \frac{1}{7} d v

Take the constant out:

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

= 20 \cdot \frac{1}{2} \cdot \frac{1}{7} \cdot \int sin (v) d v

Use the common integral:

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

= 20 \cdot \frac{1}{2} \cdot \frac{1}{7} (- cos (v))

Substitute back

= 20 \cdot \frac{1}{2} \cdot \frac{1}{7} (- cos (7 x^{2} - 8))

Simplify

20 \cdot \frac{1}{2} \cdot \frac{1}{7} (- cos (7 x^{2} - 8)) : - \frac{10}{7} cos (7 x^{2} - 8)

= - \frac{10}{7} cos (7 x^{2} - 8)

Add a constant to the solution

= - \frac{10}{7} cos (7 x^{2} - 8) + C

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What is the integral of 20xsin(7x^2-8) ?
The integral of 20xsin(7x^2-8) is -10/7 cos(7x^2-8)+C