Evaluate the Integral

Problem

(∫_^)(x√(,4−9*x2)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the expression inside the square root, 4−9*x2 is a multiple of the x term outside.

2. Substitute a new variable u for the radicand. Let:

u=4−9*x2

3. Differentiate u with respect to x to find the relationship between d(u) and d(x)

d(u)/d(x)=−18*x

d(u)=−18*x*d(x)

x*d(x)=−1/18*d(u)

4. Rewrite the integral in terms of u by substituting the expressions found in the previous steps:

(∫_^)(√(,u)*(−1/18)*d(u))

5. Factor out the constant from the integral:

−1/18*(∫_^)(u(1/2)*d(u))

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

−1/18*((u(3/2))/(3/2))+C

7. Simplify the resulting expression:

−1/18*(2/3*u(3/2))+C

−1/27*u(3/2)+C

8. Back-substitute the original expression for u to return to the variable x

−1/27*(4−9*x2)(3/2)+C

Final Answer

(∫_^)(x√(,4−9*x2)*d(x))=−1/27*(4−9*x2)(3/2)+C

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