What is $f (x) = \int x sin 3 x d x$ $f(x) = int xsin3x dx$ if $f (\frac{π}{3}) = - 2$ $f(pi/3) = -2$ ?

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What is $f (x) = \int x sin 3 x d x$ $f(x) = int xsin3x dx$ if $f (\frac{π}{3}) = - 2$ $f(pi/3) = -2$ ?

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Answer 1 · 2016-01-23T22:20:29

$f (x) = - \frac{x}{3} cos (3 x) + \frac{1}{9} sin (3 x) - \frac{π}{9}$ $f(x) = -x/3cos(3x) + 1/9sin(3x) - pi/9$

Explanation:

Use integration by parts:

$\int u d v = u v - \int v d u$ $intudv = uv - intv du$
$u = x; d v = sin (3 x) d x$ $u = x; dv = sin(3x) dx$
$d u = d x; \int d v = \int sin (3 x) d x; v = - \frac{1}{3} cos (3 x)$ $du = dx; intdv = intsin(3x) dx; v = -1/3 cos(3x)$
$\int u d v = - \frac{x}{3} cos (3 x) + \frac{1}{3} \int cos (3 x) d u$ $intudv = -x/3cos(3x) + 1/3int cos(3x) du$

$f (x) = \int u d v = - \frac{x}{3} cos (3 x) + \frac{1}{9} sin (3 x) + C$ $f(x) = intudv = -x/3cos(3x) + 1/9sin(3x) + C$

To calculate the constant, C use the fact that: $f (\frac{π}{3}) = - 2;$ $f(pi/3) = -2;$
$f(pi/3) = -pi/9cos(pi) + cancel(1/3 sin(pi)) = -pi/9$

$f(x) = -x/3cos(3x) + 1/9sin(3x) - pi/9$

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What is $f (x) = \int x sin 3 x d x$ $f(x) = int xsin3x dx$ if $f (\frac{π}{3}) = - 2$ $f(pi/3) = -2$ ?

1 Answer

Explanation:

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Explanation:

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