Find the Derivative of the Integral

Problem

d()/d(x)*(∫_^)(sin(2*x)/sin(x)*d(x))

Solution

1. Identify the operation required, which is to find the derivative of an indefinite integral with respect to its variable of integration.

2. Apply the Fundamental Theorem of Calculus, which states that the derivative of an indefinite integral of a function is the function itself.

3. State the relationship:

d()/d(x)*(∫_^)(ƒ(x)*d(x))=ƒ(x)

4. Substitute the given integrand into the formula:

ƒ(x)=sin(2*x)/sin(x)

5. Simplify the expression using the double-angle identity sin(2*x)=2*sin(x)*cos(x)

(2*sin(x)*cos(x))/sin(x)=2*cos(x)

Final Answer

d()/d(x)*(∫_^)(sin(2*x)/sin(x)*d(x))=2*cos(x)

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