Evaluate the Integral

Problem

(∫_^)((t3)/√(,1−t8)*d(t))

Solution

1. Identify a substitution that simplifies the denominator by recognizing that t8 is the square of t4

2. Substitute u=t4 which implies that the derivative is d(u)/d(t)=4*t3

3. Rearrange the differential to solve for the numerator: t3*d(t)=1/4*d(u)

4. Rewrite the integral in terms of u

(∫_^)(1/(4√(,1−u2))*d(u))

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

6. Substitute back u=t4 to express the result in terms of the original variable t

Final Answer

(∫_^)((t3)/√(,1−t8)*d(t))=1/4*arcsin(t4)+C

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