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

Integrate by parts using the formula , where and .

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.

Multiply by .

Multiply by .

Integrate by parts using the formula , where and .

Combine and .

Combine and .

Combine and .

Combine and .

Since is constant with respect to , move out of the integral.

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.

Multiply and .

Multiply by .

The integral of with respect to is .

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 .

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 .

Add and .

Multiply by .

Add and .

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 .

Add and .

Multiply by .

is a full rotation so replace with .

The exact value of is .

Multiply by .

Add and .

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 .

Add and .

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

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