Evaluate the Integral

Problem

(∫_^)((a+b*x2)/√(,3*a*x+b*x3)*d(x))

Solution

1. Identify the substitution by observing that the derivative of the expression inside the square root is a multiple of the numerator.

2. Define the substitution variable u=3*a*x+b*x3

3. Differentiate u with respect to x to find d(u)

d(u)/d(x)=3*a+3*b*x2

4. Factor the expression for d(u) to match the numerator of the integral.

d(u)=3*(a+b*x2)*d(x)

1/3*d(u)=(a+b*x2)*d(x)

5. Substitute the expressions for u and d(u) into the integral.

(∫_^)(1/(3√(,u))*d(u))

6. Rewrite the integral using power notation for easier integration.

1/3*(∫_^)(u(−1/2)*d(u))

7. Integrate using the power rule (∫_^)(un*d(u))=(u(n+1))/(n+1)

1/3⋅(u(1/2))/(1/2)+C

8. Simplify the resulting expression.

2/3√(,u)+C

9. Back-substitute the original expression for u to get the final result.

2/3√(,3*a*x+b*x3)+C

Final Answer

(∫_^)((a+b*x2)/√(,3*a*x+b*x3)*d(x))=(2√(,3*a*x+b*x3))/3+C

Want more problems? Check here!