Evaluate the Integral

Problem

(∫_^)(√(,4*x+9)*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=4*x+9 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

4. Solve for d(x) to substitute it back into the integral.

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

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

(∫_^)(√(,u)⋅1/4*d(u))

6. Factor out the constant 1/4 and rewrite the square root as a fractional exponent.

1/4*(∫_^)(u(1/2)*d(u))

7. 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

8. Simplify the resulting expression by multiplying the fractions.

1/4⋅2/3*u(3/2)+C

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

9. Back-substitute the original expression 4*x+9 for u to get the final result in terms of x

1/6*(4*x+9)(3/2)+C

Final Answer

(∫_^)(√(,4*x+9)*d(x))=1/6*(4*x+9)(3/2)+C

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