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

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

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

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

- x^{2} cos (2 x) + x sin (2 x) + \frac{3}{2} cos (2 x) + C

Solution steps

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

Rewrite using trig identities

= \int 2 cos (x) sin (x) (- 2 + 2 x^{2}) d x

Take the constant out:

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

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

Simplify

cos (x) sin (x) (- 2 + 2 x^{2}) : sin (2 x) (- 1 + x^{2})

= 2 \cdot \int sin (2 x) (- 1 + x^{2}) d x

Apply Integration By Parts

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

\int - x cos (2 x) d x = - \frac{1}{4} (2 x sin (2 x) + cos (2 x))

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

Simplify

2 (- \frac{1}{2} cos (2 x) (- 1 + x^{2}) - (- \frac{1}{4} (2 x sin (2 x) + cos (2 x)))) : - x^{2} cos (2 x) + x sin (2 x) + \frac{3}{2} cos (2 x)

= - x^{2} cos (2 x) + x sin (2 x) + \frac{3}{2} cos (2 x)

Add a constant to the solution

= - x^{2} cos (2 x) + x sin (2 x) + \frac{3}{2} cos (2 x) + C

\int \frac{2 x}{( x + 2 ) ( x - 2 ) ^{2}} d x

\int \frac{1}{( x + 2 ) ( 3 - x )} d x

\int (4 x + 3) (ln (2 x - 3)) d x

\int \frac{2 sin ( \frac{1}{x} )}{x ^{2}} d x

\int \frac{ln ( ln ( k ) )}{2 k} d k

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