Evaluate the Integral

Problem

(∫_^)(√(,4*x+5)*d(x))

Solution

1. Identify the integral as a candidate for usubstitution because the integrand is a composite function with a linear inner part.

2. Substitute u=4*x+5 to simplify the expression under the radical.

3. Differentiate u with respect to x to find the relationship between d(u) and d(x)

d(u)/d(x)=4

⇒d(x)=1/4*d(u)

4. Rewrite the integral in terms of u

(∫_^)(√(,u)⋅1/4*d(u))

⇒1/4*(∫_^)(u(1/2)*d(u))

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

1/4⋅(u(3/2))/(3/2)+C

⇒1/4⋅2/3*u(3/2)+C

6. Simplify the coefficients:

1/6*u(3/2)+C

7. Back-substitute the original expression 4*x+5 for u

1/6*(4*x+5)(3/2)+C

Final Answer

(∫_^)(√(,4*x+5)*d(x))=1/6*(4*x+5)(3/2)+C

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