Evaluate the Integral

Problem

(∫_^)((2*x)/√(,x2+2*x+26)*d(x))

Solution

1. Complete the square for the quadratic expression inside the square root.

x2+2*x+26=(x+1)2+25

2. Substitute u=x+1 to simplify the integral, which implies x=u−1 and d(x)=d(u)

(∫_^)((2*(u−1))/√(,u2+25)*d(u))

3. Distribute the numerator and split the integral into two separate parts.

(∫_^)((2*u)/√(,u2+25)*d(u))−(∫_^)(2/√(,u2+25)*d(u))

4. Evaluate the first integral using the substitution w=u2+25 where d(w)=2*u*d(u)

(∫_^)(w(−1/2)*d(w))=2*w(1/2)=2√(,u2+25)

5. Evaluate the second integral using the standard integral form (∫_^)(1/√(,u2+a2)*d(u))=ln(u+√(,u2+a2))

2*ln(u+√(,u2+25))

6. Combine the results and substitute x+1 back in for u

2√(,(x+1)2+25)−2*ln(x+1+√(,(x+1)2+25))+C

7. Simplify the expressions inside the square roots back to the original quadratic form.

2√(,x2+2*x+26)−2*ln(x+1+√(,x2+2*x+26))+C

Final Answer

(∫_^)((2*x)/√(,x2+2*x+26)*d(x))=2√(,x2+2*x+26)−2*ln(x+1+√(,x2+2*x+26))+C

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