Evaluate the Integral

Problem

(∫_^)(x√(,2*x+1)*d(x))

Solution

1. Identify the substitution method to simplify the integrand. Let u=2*x+1

2. Differentiate the substitution equation to find the relationship between d(x) and d(u)

u=2*x+1

d(u)/d(x)=2

d(x)=1/2*d(u)

3. Solve for x in terms of u to substitute the remaining x in the integrand.

u−1=2*x

x=(u−1)/2

4. Substitute the expressions for x √(,2*x+1) and d(x) into the integral.

(∫_^)((u−1)/2⋅√(,u)⋅1/2*d(u))

5. Factor out the constants and distribute √(,u) (which is u(1/2) into the binomial.

1/4*(∫_^)((u−1)*u(1/2)*d(u))

1/4*(∫_^)((u(3/2)−u(1/2))*d(u))

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

1/4*((u(5/2))/(5/2)−(u(3/2))/(3/2))+C

1/4*(2/5*u(5/2)−2/3*u(3/2))+C

7. Simplify the expression by distributing the constant 1/4

1/10*u(5/2)−1/6*u(3/2)+C

8. Back-substitute u=2*x+1 to return to the original variable.

1/10*(2*x+1)(5/2)−1/6*(2*x+1)(3/2)+C

Final Answer

(∫_^)(x√(,2*x+1)*d(x))=1/10*(2*x+1)(5/2)−1/6*(2*x+1)(3/2)+C

Want more problems? Check here!