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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

\int \frac{2}{3} x sin (3 x) d x = \frac{2}{3} (- \frac{1}{3} x cos (3 x) + \frac{1}{9} sin (3 x))

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

Simplify

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

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

Add a constant to the solution

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

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