Evaluate the Integral

Problem

(∫_1^2)((e^1/(x^4))/(x^5)*d(x))

Solution

1. Identify a suitable substitution to simplify the exponent of the natural exponential function.

2. Let u=1/(x^4) which can be rewritten as u=x^(−4)

3. Differentiate u with respect to x to find d(u)=−4*x^(−5)*d(x) which simplifies to d(u)=−4/(x^5)*d(x)

4. Rearrange the differential to solve for the terms present in the integral: −1/4*d(u)=1/(x^5)*d(x)

5. Change the limits of integration based on u=1/(x^4) When x=1 u=1/1^4=1 When x=2 u=1/2^4=1/16

6. Substitute the new variables and limits into the integral.

(∫_1^1/16)(−1/4*e^u*d(u))

7. Apply the property of integrals to reverse the limits and remove the negative sign.

1/4*(∫_1/16^1)(e^u*d(u))

8. Integrate the function e^u with respect to u

1/4*[e^u]^1_1/16

9. Evaluate the definite integral by plugging in the upper and lower limits.

1/4*(e^1−e^1/16)

Final Answer

(∫_1^2)((e^1/(x^4))/(x^5)*d(x))=(e−√(16,e))/4

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