Evaluate the Integral

Problem

(∫_^)(5*x√(3,1−x^2)*d(x))

Solution

1. Identify the integral as a candidate for usubstitution because the derivative of the inner function 1−x^2 is a multiple of the x term outside the radical.

2. Define the substitution variable u=1−x^2

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

4. Substitute the variables into the integral to rewrite it in terms of u

(∫_^)(5√(3,u)*(−1/2)*d(u))

5. Factor out the constants from the integral.

−5/2*(∫_^)(u^(1/3)*d(u))

6. Apply the power rule for integration, which states (∫_^)(u^n*d(u))=(u^(n+1))/(n+1)+C

−5/2*((u^(4/3))/(4/3))+C

7. Simplify the coefficient by multiplying by the reciprocal.

−5/2⋅3/4*u^(4/3)+C

−15/8*u^(4/3)+C

8. Back-substitute the original expression 1−x^2 for u

−15/8*(1−x^2)^(4/3)+C

Final Answer

(∫_^)(5*x√(3,1−x^2)*d(x))=−15/8*(1−x^2)^(4/3)+C

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