Evaluate the Integral

Problem

(∫_^)((z2)/√(3,1+z3)*d(z))

Solution

1. Identify a substitution to simplify the integrand by letting u be the expression inside the cube root.

u=1+z3

2. Differentiate u with respect to z to find the relationship between d(u) and d(z)

d(u)/d(z)=3*z2

d(u)=3*z2*d(z)

3. Solve for the terms present in the integral, which are z2*d(z)

1/3*d(u)=z2*d(z)

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

(∫_^)(1/(3*u(1/3))*d(u))

5. Rewrite the integrand using a negative exponent to prepare for integration.

1/3*(∫_^)(u(−1/3)*d(u))

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

1/3⋅(u(2/3))/(2/3)+C

7. Simplify the constant coefficients.

1/3⋅3/2*u(2/3)+C

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

8. Back-substitute the original expression for u to get the final answer in terms of z

1/2*(1+z3)(2/3)+C

Final Answer

(∫_^)((z2)/√(3,1+z3)*d(z))=1/2*(1+z3)(2/3)+C

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