What is the integral of (x^2+1)sin(3x) ?

The integral of (x^2+1)sin(3x) is -1/3 x^2cos(3x)+2/9 xsin(3x)-7/27 cos(3x)+C

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integral of (x^2+1)sin(3x)

\int (x^{2} + 1) sin (3 x) d x

- \frac{1}{3} x^{2} cos (3 x) + \frac{2}{9} x sin (3 x) - \frac{7}{27} cos (3 x) + C

Solution steps

\int (x^{2} + 1) sin (3 x) d x

Apply Integration By Parts

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

\int - \frac{2}{3} x cos (3 x) d x = - \frac{2}{27} (3 x sin (3 x) + cos (3 x))

= - \frac{1}{3} cos (3 x) (x^{2} + 1) - (- \frac{2}{27} (3 x sin (3 x) + cos (3 x)))

Simplify

- \frac{1}{3} cos (3 x) (x^{2} + 1) - (- \frac{2}{27} (3 x sin (3 x) + cos (3 x))) : - \frac{1}{3} x^{2} cos (3 x) + \frac{2}{9} x sin (3 x) - \frac{7}{27} cos (3 x)

= - \frac{1}{3} x^{2} cos (3 x) + \frac{2}{9} x sin (3 x) - \frac{7}{27} cos (3 x)

Add a constant to the solution

= - \frac{1}{3} x^{2} cos (3 x) + \frac{2}{9} x sin (3 x) - \frac{7}{27} cos (3 x) + C

What is the integral of (x^2+1)sin(3x) ?
The integral of (x^2+1)sin(3x) is -1/3 x^2cos(3x)+2/9 xsin(3x)-7/27 cos(3x)+C