Find the Integral x square root of 4-x^2

Problem

(∫_^)(x√(,4−x2)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function 4−x2 is a multiple of the outer factor x

2. Substitute u=4−x2

3. Differentiate u to find d(u)=−2*x*d(x) which implies x*d(x)=−1/2*d(u)

4. Rewrite the integral in terms of u

(∫_^)(√(,u)⋅(−1/2)*d(u))

5. Factor out the constant:

−1/2*(∫_^)(u(1/2)*d(u))

6. Integrate using the power rule (∫_^)(un*d(u))=(u(n+1))/(n+1)

−1/2⋅(u(3/2))/(3/2)+C

7. Simplify the expression:

−1/3*u(3/2)+C

8. Back-substitute u=4−x2 to return to the original variable.

Final Answer

(∫_^)(x√(,4−x2)*d(x))=−1/3*(4−x2)(3/2)+C

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