Evaluate the Integral

Problem

(∫_^)(e(−x4)*(−4*x3)*d(x))

Solution

1. Identify the structure of the integrand, noting that the expression −4*x3 is the derivative of the exponent −x4

2. Apply the method of usubstitution by letting u=−x4

3. Calculate the differential d(u) by differentiating u with respect to x

d(u)=−4*x3*d(x)

4. Substitute u and d(u) into the original integral to simplify the expression.

(∫_^)(eu*d(u))

5. Integrate the exponential function with respect to u

eu+C

6. Back-substitute the original expression for u to find the final result in terms of x

e(−x4)+C

Final Answer

(∫_^)(e(−x4)*(−4*x3)*d(x))=e(−x4)+C

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