Evaluate the Integral

Problem

(∫_^)((t2)/√(,1−t6)*d(t))

Solution

1. Identify a substitution that simplifies the integrand by noting that t6=(t3)2 and the derivative of t3 is 3*t2

2. Substitute u=t3 which implies d(u)=3*t2*d(t) or 1/3*d(u)=t2*d(t)

3. Rewrite the integral in terms of u

(∫_^)(1/(3√(,1−u2))*d(u))

4. Apply the standard integral formula for the inverse sine function, where (∫_^)(1/√(,1−u2)*d(u))=arcsin(u)+C

1/3*arcsin(u)+C

5. Back-substitute u=t3 to express the final result in terms of the original variable t

Final Answer

(∫_^)((t2)/√(,1−t6)*d(t))=1/3*arcsin(t3)+C

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