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(x-3)/(3-x)=2

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∫ xsin(2x)cos(3x)dx

Solution

12 (xcos(x)−sin(x)−15 xcos(5x)+125 sin(5x))+C

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∫ xsin(2x)cos(3x)dx

Use the following identity: cos(t)sin(s)=sin(s+t)+sin(s−t)2

=∫ xsin(2x+3x)+sin(2x−3x)2 dx

Take the constant out: ∫a·f(x)dx=a·∫f(x)dx

=12 · ∫ x(sin(2x+3x)+sin(2x−3x))dx

Expand x(sin(2x+3x)+sin(2x−3x)): xsin(5x)+xsin(−x)

=12 · ∫ xsin(5x)+xsin(−x)dx

Simplify xsin(5x)+xsin(−x): −xsin(x)+xsin(5x)

=12 · ∫ −xsin(x)+xsin(5x)dx

Apply the Sum Rule: ∫f(x)±g(x)dx=∫f(x)dx±∫g(x)dx

=12 (−∫ xsin(x)dx+∫ xsin(5x)dx)

∫ xsin(x)dx=−xcos(x)+sin(x)

∫ xsin(5x)dx=−15 xcos(5x)+125 sin(5x)

=12 (−(−xcos(x)+sin(x))−15 xcos(5x)+125 sin(5x))

Simplify

=12 (xcos(x)−sin(x)−15 xcos(5x)+125 sin(5x))

Add a constant to the solution

=12 (xcos(x)−sin(x)−15 xcos(5x)+125 sin(5x))+C

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−▭▭	<	7	8	9	÷	`AC`
+▭▭	>	4	5	6	×	☐☐☐
×▭▭	(	1	2	3	−	`x`
▭ \|▭	)	.	0	=	+	`y`

(x-3)/(3-x)=2

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