Find the Derivative of the Integral

Problem

d()/d(x)*(∫_^)(12*x*(x2+3)5*d(x))

Solution

1. Identify the operation required, which is to find the derivative of an indefinite integral.

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

3. State the relationship:

d()/d(x)*(∫_^)(ƒ(x)*d(x))=ƒ(x)

4. Substitute the given integrand ƒ(x)=12*x*(x2+3)5 into the theorem.

5. Conclude that the differentiation and integration operations cancel each other out.

Final Answer

d()/d(x)*(∫_^)(12*x*(x2+3)5*d(x))=12*x*(x2+3)5

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