# Evaluate integral from 0 to pi/3 of sin(3x) with respect to x

Let . Then , so . Rewrite using and .
Let . Find .
Rewrite.
Divide by .
Substitute the lower limit in for in .
Multiply by .
Substitute the upper limit in for in .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
The values found for and will be used to evaluate the definite integral.
Rewrite the problem using , , and the new limits of integration.
Combine and .
Since is constant with respect to , move out of the integral.
The integral of with respect to is .
Combine and .
Evaluate at and at .
Simplify.
The exact value of is .
Factor out of .
Rewrite as .
Factor out of .
Rewrite as .
Move the negative in front of the fraction.
Simplify the numerator.
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.
The exact value of is .
Multiply by .
Subtract from .
Move the negative in front of the fraction.
Multiply .
Multiply by .
Multiply by .
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Evaluate integral from 0 to pi/3 of sin(3x) with respect to x