Evaluate the Integral

Problem

(∫_^)(x√(3,6*x+7)*d(x))

Solution

1. Identify the substitution to simplify the radical. Let u=√(3,6*x+7)

2. Solve for x by cubing both sides: u3=6*x+7 which gives x=(u3−7)/6

3. Differentiate x with respect to u to find d(x) d(x)/d(u)=(3*u2)/6=(u2)/2 so d(x)=(u2)/2*d(u)

4. Substitute the expressions for x √(3,6*x+7) and d(x) into the integral:

(∫_^)((u3−7)/6⋅u⋅(u2)/2*d(u))

5. Simplify the integrand by pulling out the constant factor 1/12

1/12*(∫_^)((u3−7)*u3*d(u))

6. Distribute the u3 term:

1/12*(∫_^)((u6−7*u3)*d(u))

7. Integrate term by term using the power rule:

1/12*((u7)/7−(7*u4)/4)+C

8. Distribute the constant 1/12

(u7)/84−(7*u4)/48+C

9. Back-substitute u=(6*x+7)(1/3) to return to the variable x

((6*x+7)(7/3))/84−(7*(6*x+7)(4/3))/48+C

Final Answer

(∫_^)(x√(3,6*x+7)*d(x))=((6*x+7)(7/3))/84−(7*(6*x+7)(4/3))/48+C

Want more problems? Check here!