Evaluate the Integral

Problem

(∫_0^6)(x/√(,9+2*x2)*d(x))

Solution

1. Identify the substitution method to simplify the integrand. Let u=9+2*x2

2. Differentiate u with respect to x to find d(u)

d(u)/d(x)=4*x

d(u)=4*x*d(x)

1/4*d(u)=x*d(x)

3. Change the limits of integration from x to u

Lower limit: *x=0⇒u=9+2*(0)2=9

Upper limit: *x=6⇒u=9+2*(6)2=9+72=81

4. Substitute the variables and limits into the integral.

(∫_9^81)(1/(4√(,u))*d(u))

1/4*(∫_9^81)(u(−1/2)*d(u))

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

1/4*[2*u(1/2)]819

1/2*[√(,u)]819

6. Evaluate the definite integral at the boundaries.

1/2*(√(,81)−√(,9))

1/2*(9−3)

1/2*(6)=3

Final Answer

(∫_0^6)(x/√(,9+2*x2)*d(x))=3

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