Evaluate the Integral

Problem

(∫_^)(√(,8*x+1)*d(x))

Solution

1. Identify the integral as a candidate for usubstitution because the integrand is a composition of a square root function and a linear function.

2. Substitute u=8*x+1 to simplify the expression.

3. Differentiate u with respect to x to find d(u)=8*d(x) which implies d(x)=1/8*d(u)

4. Rewrite the integral in terms of u by substituting the expressions for the radicand and d(x)

(∫_^)(√(,u)⋅1/8*d(u))

5. Factor out the constant 1/8 from the integral.

1/8*(∫_^)(u(1/2)*d(u))

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

1/8⋅(u(3/2))/(3/2)+C

7. Simplify the coefficient by multiplying 1/8 by 2/3

1/12*u(3/2)+C

8. Back-substitute the original expression 8*x+1 for u to get the final result.

Final Answer

(∫_^)(√(,8*x+1)*d(x))=1/12*(8*x+1)(3/2)+C

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