What is the integral of (x+7)^2sin(x) ?

The integral of (x+7)^2sin(x) is -x^2cos(x)+2xsin(x)-14xcos(x)-47cos(x)+14sin(x)+C

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integral of (x+7)^2sin(x)

\int (x + 7)^{2} sin (x) d x

- x^{2} cos (x) + 2 x sin (x) - 14 x cos (x) - 47 cos (x) + 14 sin (x) + C

Solution steps

\int (x + 7)^{2} sin (x) d x

x + 7 = 7 (\frac{x}{7} + 1)

= \int (7 (\frac{x}{7} + 1))^{2} sin (x) d x

Simplify

(7 (\frac{x}{7} + 1))^{2} sin (x) : x^{2} sin (x) + 14 x sin (x) + 49 sin (x)

= \int x^{2} sin (x) + 14 x sin (x) + 49 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 14 x sin (x) d x + \int 49 sin (x) d x

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

\int 14 x sin (x) d x = 14 (- x cos (x) + sin (x))

\int 49 sin (x) d x = - 49 cos (x)

= - x^{2} cos (x) + 2 (x sin (x) + cos (x)) + 14 (- x cos (x) + sin (x)) - 49 cos (x)

Simplify

- x^{2} cos (x) + 2 (x sin (x) + cos (x)) + 14 (- x cos (x) + sin (x)) - 49 cos (x) : - x^{2} cos (x) + 2 x sin (x) - 14 x cos (x) - 47 cos (x) + 14 sin (x)

= - x^{2} cos (x) + 2 x sin (x) - 14 x cos (x) - 47 cos (x) + 14 sin (x)

Add a constant to the solution

= - x^{2} cos (x) + 2 x sin (x) - 14 x cos (x) - 47 cos (x) + 14 sin (x) + C

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\int \frac{1}{2 x + a} d x

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What is the integral of (x+7)^2sin(x) ?
The integral of (x+7)^2sin(x) is -x^2cos(x)+2xsin(x)-14xcos(x)-47cos(x)+14sin(x)+C