What is the integral of (x^2+4)cos(5x) ?

The integral of (x^2+4)cos(5x) is 1/5 x^2sin(5x)+2/25 xcos(5x)+98/125 sin(5x)+C

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integral of (x^2+4)cos(5x)

\int (x^{2} + 4) cos (5 x) d x

\frac{1}{5} x^{2} sin (5 x) + \frac{2}{25} x cos (5 x) + \frac{98}{125} sin (5 x) + C

Solution steps

\int (x^{2} + 4) cos (5 x) d x

Expand

(x^{2} + 4) cos (5 x) : x^{2} cos (5 x) + 4 cos (5 x)

= \int x^{2} cos (5 x) + 4 cos (5 x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{2} cos (5 x) d x + \int 4 cos (5 x) d x

\int x^{2} cos (5 x) d x = \frac{1}{125} (25 x^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x)))

\int 4 cos (5 x) d x = \frac{4}{5} sin (5 x)

= \frac{1}{125} (25 x^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x))) + \frac{4}{5} sin (5 x)

Simplify

\frac{1}{125} (25 x^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x))) + \frac{4}{5} sin (5 x) : \frac{1}{5} x^{2} sin (5 x) + \frac{2}{25} x cos (5 x) + \frac{98}{125} sin (5 x)

= \frac{1}{5} x^{2} sin (5 x) + \frac{2}{25} x cos (5 x) + \frac{98}{125} sin (5 x)

Add a constant to the solution

= \frac{1}{5} x^{2} sin (5 x) + \frac{2}{25} x cos (5 x) + \frac{98}{125} sin (5 x) + C

What is the integral of (x^2+4)cos(5x) ?
The integral of (x^2+4)cos(5x) is 1/5 x^2sin(5x)+2/25 xcos(5x)+98/125 sin(5x)+C