What is #f(x) = int xsin3x dx# if #f(pi/3) = -2 #?

1 Answer
Yonas Yohannes
Jan 23, 2016

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

Explanation:

Use integration by parts:

#intudv = uv - intv du #
#u = x; dv = sin(3x) dx #
#du = dx; intdv = intsin(3x) dx; v = -1/3 cos(3x) #
#intudv = -x/3cos(3x) + 1/3int cos(3x) du #

#f(x) = intudv = -x/3cos(3x) + 1/9sin(3x) + C #

To calculate the constant, C use the fact that: #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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