What is the integral of (x+9)^2sin(x) ?

The integral of (x+9)^2sin(x) is -x^2cos(x)+2xsin(x)-18xcos(x)-79cos(x)+18sin(x)+C

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integral of (x+9)^2sin(x)

\int (x + 9)^{2} sin (x) d x

- x^{2} cos (x) + 2 x sin (x) - 18 x cos (x) - 79 cos (x) + 18 sin (x) + C

Solution steps

\int (x + 9)^{2} sin (x) d x

x + 9 = 9 (\frac{x}{9} + 1)

= \int (9 (\frac{x}{9} + 1))^{2} sin (x) d x

Simplify

(9 (\frac{x}{9} + 1))^{2} sin (x) : x^{2} sin (x) + 18 x sin (x) + 81 sin (x)

= \int x^{2} sin (x) + 18 x sin (x) + 81 sin (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} sin (x) d x + \int 18 x sin (x) d x + \int 81 sin (x) d x

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

\int 18 x sin (x) d x = 18 (- x cos (x) + sin (x))

\int 81 sin (x) d x = - 81 cos (x)

= - x^{2} cos (x) + 2 (x sin (x) + cos (x)) + 18 (- x cos (x) + sin (x)) - 81 cos (x)

Simplify

- x^{2} cos (x) + 2 (x sin (x) + cos (x)) + 18 (- x cos (x) + sin (x)) - 81 cos (x) : - x^{2} cos (x) + 2 x sin (x) - 18 x cos (x) - 79 cos (x) + 18 sin (x)

= - x^{2} cos (x) + 2 x sin (x) - 18 x cos (x) - 79 cos (x) + 18 sin (x)

Add a constant to the solution

= - x^{2} cos (x) + 2 x sin (x) - 18 x cos (x) - 79 cos (x) + 18 sin (x) + C

\int (2 tan (3 x) - sec (2 x)) d x

\int \frac{x ^{2} - a ^{2}}{4 x} d x

\int y^{3} + 3 y d y

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

\int \frac{y}{y ^{3} + 4} d y

What is the integral of (x+9)^2sin(x) ?
The integral of (x+9)^2sin(x) is -x^2cos(x)+2xsin(x)-18xcos(x)-79cos(x)+18sin(x)+C