Evaluate the Integral

Problem

(∫_0^√(,3))((4*x)/√(,x2+1)*d(x))

Solution

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

2. Differentiate the substitution to find d(u) Since u=x2+1 then d(u)=2*x*d(x)

3. Rewrite the integral in terms of u Notice that 4*x*d(x)=2*(2*x*d(x))=2*d(u)

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

5. Substitute the new variables and limits into the integral.

(∫_1^4)(2/√(,u)*d(u))

6. Rewrite the integrand using a power for easier integration.

(∫_1^4)(2*u(−1/2)*d(u))

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

[2⋅(u(1/2))/(1/2)]41

8. Simplify the expression before evaluating.

[4√(,u)]41

9. Evaluate at the upper and lower limits.

4√(,4)−4√(,1)

10. Calculate the final numerical value.

8−4=4

Final Answer

(∫_0^√(,3))((4*x)/√(,x2+1)*d(x))=4

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