Evaluate the Integral

Problem

(∫_^)(20*t3*e(−t4)*d(t))

Solution

1. Identify a suitable substitution to simplify the integrand, noting that the derivative of the exponent −t4 is proportional to the t3 term outside.

2. Substitute u=−t4

3. Differentiate u with respect to t to find d(u)=−4*t3*d(t)

4. Rearrange the differential to solve for the terms in the integral: −1/4*d(u)=t3*d(t)

5. Rewrite the integral in terms of u by substituting the expressions found: (∫_^)(20*eu*(−1/4)*d(u))

6. Simplify the constant coefficients: (∫_^)(−5*eu*d(u))

7. Integrate the exponential function: −5*eu+C

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

Final Answer

(∫_^)(20*t3*e(−t4)*d(t))=−5*e(−t4)+C

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