# Evaluate integral from zero to four pi of t to the power of two sin(two t) with respect to t

Evaluate integral from 0 to 4pi of t^2sin(2t) with respect to t
Integrate by parts using the formula , where and .
Simplify.
Combine and .
Combine and .
Combine and .
Multiply by .
Combine and .
Combine and .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Since is constant with respect to , move out of the integral.
Simplify.
Multiply by .
Multiply by .
Integrate by parts using the formula , where and .
Simplify.
Combine and .
Combine and .
Combine and .
Combine and .
Since is constant with respect to , move out of the integral.
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 .
Multiply by .
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.
Simplify.
Multiply and .
Multiply by .
The integral of with respect to is .
Simplify.
Combine and .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Move to the left of .
Substitute and simplify.
Evaluate at and at .
Evaluate at and at .
Simplify.
Factor out of .
Apply the product rule to .
Raise to the power of .
Multiply by .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by .
Multiply by .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Raising to any positive power yields .
Multiply by .
Multiply by .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by .
Multiply by .
Multiply by .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by .
The exact value of is .
is a full rotation so replace with .
The exact value of is .
Multiply by .
is a full rotation so replace with .
The exact value of is .
Multiply .
Multiply by .
Multiply by .
Multiply by .
is a full rotation so replace with .
The exact value of is .
Multiply by .
Multiply by .
Reduce the expression by cancelling the common factors.
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .