Evaluate the Integral

Problem

(∫_^)((t^2)/√(,1−t^6)*d(t))

Solution

1. Identify a substitution that simplifies the integrand by noting that t^6=(t^3)^2 and the derivative of t^3 is 3*t^2

2. Substitute u=t^3 which implies d(u)=3*t^2*d(t) or 1/3*d(u)=t^2*d(t)

3. Rewrite the integral in terms of u

(∫_^)(1/(3√(,1−u^2))*d(u))

4. Apply the standard integral formula for the inverse sine function, where (∫_^)(1/√(,1−u^2)*d(u))=arcsin(u)+C

1/3*arcsin(u)+C

5. Back-substitute u=t^3 to express the final result in terms of the original variable t

Final Answer

(∫_^)((t^2)/√(,1−t^6)*d(t))=1/3*arcsin(t^3)+C

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