Evaluate the Integral

Problem

(∫_^)(3*t^2√(,t^3+2)*d(t))

Solution

1. Identify the substitution by looking for a function and its derivative within the integrand.

2. Substitute u=t^3+2 which implies that the derivative is d(u)/d(t)=3*t^2

3. Rewrite the differential d(t) in terms of d(u) as d(u)=3*t^2*d(t)

4. Transform the integral into the uvariable by replacing t^3+2 with u and 3*t^2*d(t) with d(u)

(∫_^)(√(,u)*d(u))

5. Apply the power rule for integration, (∫_^)(u^n*d(u))=(u^(n+1))/(n+1)+C where n=1/2

(u^(3/2))/(3/2)+C

6. Simplify the coefficient by multiplying by the reciprocal.

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

7. Back-substitute the original expression t^3+2 for u to get the final result.

Final Answer

(∫_^)(3*t^2√(,t^3+2)*d(t))=(2*(t^3+2)^(3/2))/3+C

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