Evaluate the Integral

Problem

(∫_0^π/8)(sin(2*x)*d(x))

Solution

1. Identify the integral as a definite integral of a trigonometric function requiring a simple substitution or the reverse chain rule.

2. Find the antiderivative of sin(2*x) by using the rule (∫_^)(sin(a*x)*d(x))=−1/a*cos(a*x)

(∫_^)(sin(2*x)*d(x))=−1/2*cos(2*x)

3. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit π/8 and the lower limit 0

[−1/2*cos(2*x)]π/80

4. Substitute the upper limit into the expression.

−1/2*cos(2⋅π/8)=−1/2*cos(π/4)

5. Substitute the lower limit into the expression.

−1/2*cos(2⋅0)=−1/2*cos(0)

6. Subtract the lower limit evaluation from the upper limit evaluation.

(−1/2*cos(π/4))−(−1/2*cos(0))

7. Evaluate the trigonometric values cos(π/4)=√(,2)/2 and cos(0)=1

−1/2⋅√(,2)/2+1/2⋅1

8. Simplify the resulting numerical expression.

−√(,2)/4+2/4=(2−√(,2))/4

Final Answer

(∫_0^π/8)(sin(2*x)*d(x))=(2−√(,2))/4

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