Evaluate the Integral

Problem

(∫_^)(((√(,x)+2)2)/(5√(,x))*d(x))

Solution

1. Expand the numerator using the square of a binomial formula (a+b)2=a2+2*a*b+b2

(√(,x)+2)2=x+4√(,x)+4

2. Rewrite the integral by substituting the expanded numerator and factoring out the constant 1/5

1/5*(∫_^)((x+4√(,x)+4)/√(,x)*d(x))

3. Simplify the integrand by dividing each term in the numerator by √(,x) which is x(1/2)

1/5*(∫_^)((x(1/2)+4+4*x(−1/2))*d(x))

4. Apply the power rule for integration (∫_^)(xn*d(x))=(x(n+1))/(n+1) to each term.

1/5*((x(3/2))/(3/2)+4*x+(4*x(1/2))/(1/2))+C

5. Simplify the coefficients within the parentheses.

1/5*(2/3*x(3/2)+4*x+8*x(1/2))+C

6. Distribute the 1/5 to obtain the final expression.

2/15*x(3/2)+4/5*x+8/5*x(1/2)+C

Final Answer

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

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