Evaluate the Integral

Problem

(∫_0^√(,3))((6*x)/√(,x2+1)*d(x))

Solution

1. Identify the substitution method to simplify the integrand. Let u=x2+1

2. Calculate the differential d(u) by differentiating u with respect to x which gives d(u)=2*x*d(x)

3. Rewrite the numerator 6*x*d(x) in terms of d(u) as 3*(2*x*d(x))=3*d(u)

4. Determine the new limits of integration for u When x=0 u=0+1=1 When x=√(,3) u=(√(,3))2+1=4

5. Substitute the variables and limits into the integral to get (∫_1^4)(3/√(,u)*d(u))

6. Rewrite the integrand using a power of u as 3*u(−1/2)

7. Integrate using the power rule (∫_^)(un*d(u))=(u(n+1))/(n+1) This results in 3⋅(u(1/2))/(1/2)=6√(,u)

8. Evaluate the definite integral at the boundaries u=4 and u=1

9. Calculate the final value as 6√(,4)−6√(,1)=6*(2)−6*(1)=12−6=6

Final Answer

(∫_0^√(,3))((6*x)/√(,x2+1)*d(x))=6

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