Evaluate the Integral

Problem

(∫_0^π/2)(cos((2*x)/3)*d(x))

Solution

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

2. Apply the substitution u=(2*x)/3 which implies d(u)=2/3*d(x) or d(x)=3/2*d(u)

3. Change the limits of integration: when x=0 u=0 when x=π/2 u=π/3

4. Integrate the function cos(u) with respect to u

(∫_^)(cos(u)*d(u))=sin(u)

5. Evaluate the antiderivative 3/2*sin((2*x)/3) at the upper and lower limits.

[3/2*sin((2*x)/3)](π/2)0

6. Substitute the upper limit π/2 into the expression.

3/2*sin((2*(π/2))/3)=3/2*sin(π/3)

7. Substitute the lower limit 0 into the expression.

3/2*sin(0)=0

8. Calculate the final value using the known value sin(π/3)=√(,3)/2

3/2⋅√(,3)/2=(3√(,3))/4

Final Answer

(∫_0^π/2)(cos((2*x)/3)*d(x))=(3√(,3))/4

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