Evaluate the Integral integral of x^4e^(x^5) with respect to x

Problem

(∫_^)(x4*e(x5)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the exponent x5 is a multiple of the x4 term present in the integrand.

2. Substitute u=x5 to simplify the expression.

3. Differentiate u with respect to x to find the relationship between d(u) and d(x)

d(u)/d(x)=5*x4

d(u)=5*x4*d(x)

4. Isolate the terms present in the integral by dividing both sides by 5

1/5*d(u)=x4*d(x)

5. Rewrite the integral in terms of u by substituting x5 with u and x4*d(x) with 1/5*d(u)

(∫_^)(1/5*eu*d(u))

6. Integrate the exponential function with respect to u

1/5*eu+C

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

1/5*e(x5)+C

Final Answer

(∫_^)(x4*e(x5)*d(x))=(e(x5))/5+C

Want more problems? Check here!