Evaluate the Integral integral of x^3e^(x^4) with respect to x

Problem

(∫_^)(x3*e(x4)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the exponent x4 is a multiple of the x3 term present in the integrand.

2. Substitute u=x4 to simplify the expression.

3. Differentiate u with respect to x to find d(u)=4*x3*d(x)

4. Rearrange the differential to solve for the terms in the integral: 1/4*d(u)=x3*d(x)

5. Rewrite the integral in terms of u

(∫_^)(1/4*eu*d(u))

6. Integrate the exponential function:

1/4*eu+C

7. Back-substitute x4 for u to return to the original variable.

Final Answer

(∫_^)(x3*e(x4)*d(x))=(e(x4))/4+C

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